# A Simple Nonconforming Tetrahedral Element for the Stokes Equations

Peter Hansbo  Department of Mechanical Engineering, Jönköping University,
SE-55111 Jönköping, Sweden.  Mats G. Larson  Department of Mathematics and
Mathematical Statistics, Umeå Univerity, SE-90187 Umeå, Sweden.

###### Abstract

In this paper we apply a nonconforming rotated bilinear tetrahedral element to
the Stokes problem in ${\mathbb{R}}^{3}$. We show that the element is stable
in combination with a piecewise linear, continuous, approximation of the
pressure. This gives an approximation similar to the well known continuous
$P^{2}$–$P^{1}$ Taylor–Hood element, but with fewer degrees of freedom. The
element is a stable non–conforming low order element which fulfils Korn’s
inequality, leading to stability also in the case where the Stokes equations
are written on stress form for use in the case of free surface flow.

###### keywords:

Finite element method, nonconforming element, Stokes equations.

##  1 INTRODUCTION

The nonconforming rotated $Q_{1}$ tetrahedron is derived from the
nonconforming hexahedral element proposed by Rannacher and Turek [6] and was
applied to linear and nonlinear elasticity problems in [3, 4, 5]. It has
properties similar to the hexahedral element, with improved bending behaviour
in elasticity, compared to the $P^{1}$ tetrahedron, and it allows for diagonal
mass lumping for explicit time–stepping in dynamic problems.

In this paper, we investigate its properties as a Stokes element, and show
that in combination with linear, continuous pressures, it is inf–sup stable.
In a sense it is thus a reduced Taylor–Hood [7] element with fewer degrees of
freedom. It is one of the lowest order elements that is stable for Stokes,
and, unlike some other low order non–conforming elements [2, 6], it fulfills
Korn’s inequality and can thus handle the strain form of Stokes, cf. [3].

An outline of the paper is as follows: in Section 2 we recall the rotated
$Q_{1}$ element; in Section 3 we apply it to the Stokes equations and prove
stability and convergence; in Section 4 we give some numerical examples to
show the properties of the approximation.

##  2 THE ROTATED $Q_{1}$ APPROXIMATION

In order to define a low order approximation which is stable for the Stokes
problem, Rannacher and Turek [6] constructed a hexahedral element with nodes
on the faces. Two different kinds of continuity can now be imposed: point-wise
continuity and average continuity and we will in this paper consider point-
wise continuity. The Rannacher-Turek element may be viewed as an extension of
the classical Crouzeix-Raviart nonconforming tetrahedral element [2], for
which point-wise and average continuity is identical.

The difficulty in the construction of the Rannacher-Turek element compared to
the Crouzeix-Raviart element is that we have six degrees of freedom while the
dimension of the space of linears is four, trilinears eight, and quadratics
ten. Therefore we shall start with linears and add two suitable quadratic
functions in such a way that the nodal mapping is invertible. To that end let
$\\{{\boldsymbol{e}}_{i}\\}_{i=1}^{3}$ be an orthonormal coordinate system in
${\mathbb{R}}^{3}$ and let $\hat{K}=[-1,1]^{3}$ be the reference cube. The
midpoints of the faces of $\hat{K}$ takes the form
$\\{{\boldsymbol{x}}_{i}\\}_{i=1}^{6}=\\{\pm{\boldsymbol{e}}_{i}\\}_{i=1}^{3}$,
and therefore has one coordinate equal to $\pm 1$ and the other two equal to
$0$. See Fig. 1 for the enumeration of the midpoints. We now seek a space
${V}(\hat{K})$ of shape functions on $\hat{K}$ such that
${P}_{1}(\hat{K})\subset{V}(\hat{K})\subset P_{2}(\hat{K})$ and that has the
values at the six midpoints $\\{x_{i}\\}_{i=1}^{6}$ of the faces as degrees of
freedom. The linear functions
${P}_{1}(\hat{K})=\text{span}(1,x_{1},x_{2},x_{3})$ is a four dimensional
vector space and therefore we need to add two quadratic polynomials to obtain
a six dimensional space. To find these quadratic polynomials let
$N:V(\hat{K})\ni
v\mapsto[v({\boldsymbol{x}}_{i})]_{i=1}^{6}\in{\mathbb{R}}^{6}$ be the nodal
mapping and note that $N(x_{i}x_{j})={\boldsymbol{0}}$, for $i\neq j$, and
therefore we are restricted to adding polynomials in
$\text{span}(x_{1}^{2},x_{2}^{2},x_{3}^{2})$. Now
$N(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})=N(1)$ and it is therefore natural to
consider the two dimensional space

$R_{2}(\hat{K})=\\{v\in\text{span}(x_{1}^{2},x_{2}^{2},x_{3}^{2}):N(v)\cdot
N(1)=0\\}=\text{span}(x_{1}^{2}-x_{2}^{2},x_{2}^{2}-x_{3}^{2})$ (1)

where the two (non unique) basis functions on the right hand side is easily
chosen by observing that $N(b_{1}x_{1}^{2}+b_{2}x_{2}^{2}+b_{3}x_{3}^{2})\cdot
N(1)=2(b_{1}+b_{2}+b_{3})$. We define

$\boxed{V(\hat{K})=P_{1}(\hat{K})+R_{2}(\hat{K})=\text{span}(1,x_{1},x_{2},x_{3},x_{1}^{2}-x_{2}^{2},x_{2}^{2}-x_{3}^{2})}$
(2)

and verify by explicit calculation that the coordinate mapping
$N:V(\hat{K})\rightarrow{\mathbb{R}}^{6}$ is indeed invertible. Solving for
Lagrange basis functions $\varphi_{i}$ such that
$N(\varphi_{i})=[\delta_{ij}]_{j=1}^{6}$, gives

$V(\hat{K})=\text{span}(\varphi_{i})_{i=1}^{6}$ (3)

where

$\begin{array}[]{>{\displaystyle}l}\varphi_{1}=\varphi(x_{1},x_{2},x_{3})\\\\[8.53581pt]
\varphi_{2}=\varphi(-x_{1},x_{2},x_{3})\\\\[8.53581pt]
\varphi_{3}=\varphi(x_{2},x_{1},x_{3})\\\\[8.53581pt]
\varphi_{4}=\varphi(-x_{2},x_{1},x_{3})\\\\[8.53581pt]
\varphi_{5}=\varphi(x_{3},x_{1},x_{2})\\\\[8.53581pt]
\varphi_{6}=\varphi(-x_{3},x_{1},x_{2})\end{array}$ (4)

and

$\varphi(x_{1},x_{2},x_{3})=\frac{1}{6}(1+3x_{1}+2x_{1}^{2}-x_{2}^{2}-x_{3}^{2})$
(5)

The terminology rotated $Q_{1}$ elements is motivated by the fact that
$x_{1}^{2}-x_{2}^{2}=(x_{1}-x_{2})(x_{1}+x_{2})=\xi_{1}\xi_{2}$ where
$\xi_{1}=x_{1}-x_{2}$ and $\xi_{1}+\xi_{2}$ are the degrees of freedom in a
coordinate system rotated $\pi/2$ around the ${\boldsymbol{e}}_{3}$ axis.

In [3], we made the observation that there is a reference tetrahedron
$\hat{T}$ inscribed in the reference hexahedron $\hat{K}$, with edges that are
diagonals of the faces of $\hat{K}$, and thus the midpoints of the faces of
$\hat{K}$ are precisely the midpoints of the edges of $\hat{T}$, see Fig. 1.
We note that for the reference element all edges have the same length and the
centre of gravity is the origin. Using an affine map $F:\hat{T}\rightarrow T$,
we map mid side nodes in the reference configuration to mid edge nodes in the
physical configuration.

Let $\mathcal{T}_{h}:=\\{T\\}$ be a conforming, shape regular tetrahedrization
of $\Omega\subset{\mathbb{R}}^{3}$ with mesh parameter $h\in(0,h_{0}]$. We
also let $\mathcal{F}_{h}$ be the set of faces and $\mathcal{E}_{h}$ the set
of edges. We make the standard assumption for the Taylor–Hood approximation
[1], that

every $T\in\mathcal{T}_{h}$ has at least three internal edges (6)

We define the non-conforming finite element space

$\boxed{V_{h}:=\\{v\in L^{2}(\Omega):v|_{T}\circ F\in
V(\hat{T})\;\;T\in\mathcal{T}_{h},\text{$v$ is continuous in the midpoints of
$E\in\mathcal{E}_{h,I}$}\\}}$ (7)

with midpoint continuity for all interior edges. We note that
$\text{dim}(V_{h})=|\mathcal{E}_{h}|$, the number of edges in
$\mathcal{T}_{h}$.

##  3 APPLICATION TO THE STOKES EQUATIONS

###  3.1 Problem Formulation and Finite Element Approximation

We consider the Stokes equations in a domain $\Omega$ in ${\mathbb{R}}^{3}$:
find the velocity ${\boldsymbol{u}}=\left[u_{i}\right]_{i=1}^{3}$ and the
pressure $p$ such that

$\left\\{\begin{array}[]{rcl}-\Delta{\boldsymbol{u}}+\nabla
p&=&{\boldsymbol{f}}\quad\mbox{in $\Omega$}\\\
\nabla\cdot{\boldsymbol{u}}&=&\textbf{0}\quad\mbox{in $\Omega$}\\\
{\boldsymbol{u}}&=&\textbf{0}\quad\mbox{on
$\partial\Omega$}\end{array}\right.$ (8)

Let us define the spaces

$W:=\\{{\boldsymbol{v}}:{\boldsymbol{v}}\in[H^{1}(\Omega)]^{3},\;{\boldsymbol{v}}\;\text{is
zero on $\partial\Omega$}\\}$ (9)

and

$Q:=\\{q:q\in L^{2}(\Omega),\;(q,1)_{\Omega}=0\\}$ (10)

where $(\cdot,\cdot)_{\Omega}$ is the standard $L^{2}$ scalar product. Then we
have the weak form of (8): find $({\boldsymbol{u}},p)\in W\times Q$ such that

$\boxed{a({\boldsymbol{u}},{\boldsymbol{v}})-b({\boldsymbol{v}},p)+b({\boldsymbol{u}},q)=({\boldsymbol{f}},{\boldsymbol{v}})\quad\forall({\boldsymbol{v}},q)\in
W\times Q}$ (11)

where the forms are given by

$a({\boldsymbol{v}},{\boldsymbol{w}}):=(\nabla{\boldsymbol{v}},\nabla{\boldsymbol{w}})_{\Omega},\quad
b({\boldsymbol{v}},q):=(\nabla\cdot{\boldsymbol{v}},q)_{\Omega}$ (12)

To define the finite element method, we introduce the non-conforming finite
element space constructed from the space $V_{h}$ in (7) by defining

$W_{h}:=\\{{\boldsymbol{v}}:{\boldsymbol{v}}\in[V_{h}]^{3},\;{\boldsymbol{v}}\;\text{is
zero in the midpoints of $E\in\mathcal{E}_{h}$ on $\partial\Omega$}\\}$ (13)

and the space of continuous piecewise linear polynomials

$Q_{h}:=\\{q:q\in C^{0}(\Omega)\cap Q,\;q\;\text{is linear on $T$, $\forall
T\in\mathcal{T}_{h}$}\\}$ (14)

The finite element method is to find $({\boldsymbol{u}}_{h},p_{h})\in
W_{h}\times Q_{h}$ such that

$\displaystyle
A_{h}({\boldsymbol{u}}_{h},p_{h}),({\boldsymbol{v}},q))=({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}\qquad\forall({\boldsymbol{v}},q)\in
W_{h}\times Q_{h}$ (15)

Here the form is defined by

$A_{h}(({\boldsymbol{v}},q),({\boldsymbol{w}},r))=a_{h}({\boldsymbol{v}},{\boldsymbol{w}})-b({\boldsymbol{v}},r)+b({\boldsymbol{w}},p)$
(16)

with

$a_{h}({\boldsymbol{v}},{\boldsymbol{w}}):=\sum_{T\in\mathcal{T}_{h}}(\nabla{\boldsymbol{v}},\nabla{\boldsymbol{w}})_{T},\quad\quad
b({\boldsymbol{v}},q):=\sum_{T\in\mathcal{T}_{h}}(\nabla\cdot{\boldsymbol{v}},q)_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{v}},q)_{\partial
T}=\sum_{T\in\mathcal{T}_{h}}({\boldsymbol{v}},\nabla q)_{T}$ (17)

and $\nabla{\boldsymbol{v}}={\boldsymbol{v}}\otimes\nabla$ is the tensor with
elements $({\boldsymbol{v}}\otimes\nabla)_{ij}=\partial_{j}v_{i}$.

###### Remark 3.1

We will also consider the alternative form

$\displaystyle\tilde{b}({\boldsymbol{v}},q)=\sum_{T\in\mathcal{T}_{h}}(\nabla\cdot{\boldsymbol{v}},q)_{T}=\sum_{T\in\mathcal{T}_{h}}-({\boldsymbol{v}},\nabla
q)_{T}+({\boldsymbol{n}}\cdot{\boldsymbol{v}},q)_{\partial T}$ (18)

This form is preferable in the presence of natural boundary conditions, and
requires less numerical computations, but we will see that the proof of the
inf-sup condition is more complicated due to the presence of the trace term on
$\partial T$ in the right hand side of (18). Throughout the paper we will
focus our presentation on the form $b$ and we will for each result add a
remark on the modifications necessary to obtain the corresponding result for
$\tilde{b}$. Finally, we will present an inf-sup result for $\tilde{b}$ in A.

###### Remark 3.2

Unlike some nonconforming approximations, the rotated $Q_{1}$ approximation
fulfills Korn’s inequality [3], which means that we may also use the strain
form of Stokes: find the velocity ${\boldsymbol{u}}$ and the pressure $p$ such
that

$\left\\{\begin{array}[]{rcl}-2{\boldsymbol{\nabla}}\cdot{\boldsymbol{\varepsilon}}({\boldsymbol{u}})+\nabla
p&=&{\boldsymbol{f}}\quad\mbox{in $\Omega$}\\\
\nabla\cdot{\boldsymbol{u}}&=&\text{\bf 0}\quad\mbox{in $\Omega$}\\\
{\boldsymbol{u}}&=&\text{\bf 0}\quad\text{on
$\partial\Omega$}\end{array}\right.$ (19)

where

$({\boldsymbol{\varepsilon}})_{ij}:=\frac{1}{2}\left(\frac{\partial
u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)$ (20)

is the strain tensor and ${\boldsymbol{\nabla}}\cdot$ denotes matrix
divergence. This is of interest in free surface flows where we need zero
stress as a natural boundary condition, cf. Section 4.2.

###  3.2 Norms and Continuity of the Forms

Define the norms

$\|{\boldsymbol{v}}\|_{a_{h}}:=a_{h}({\boldsymbol{v}},{\boldsymbol{v}})^{1/2},\quad\|q\|_{\Omega}:=\|q\|_{L^{2}(\Omega)}$
(21)

and

$\displaystyle|||({\boldsymbol{v}},q)|||_{h}^{2}=\|{\boldsymbol{v}}\|^{2}_{a_{h}}+\|q\|^{2}_{\Omega}+h^{2}\|\nabla
q\|^{2}_{\Omega}$ (22)

We let $a\lesssim b$ denote $a\leq Cb$ with $C$ a positive constant
independent of the mesh parameter. Then we have the following continuities of
the forms $b$ and $A_{h}$.

###### Lemma 3.1

There are constants such that for all functions in $(W+W_{h})\times Q$,

$\displaystyle\boxed{b({\boldsymbol{v}},q)\lesssim\|\nabla{\boldsymbol{v}}\|_{\Omega}\Big{(}\|q\|_{\Omega}+h\|\nabla
q\|_{\Omega}\Big{)}}$ (23)

and

$\displaystyle\boxed{A_{h}(({\boldsymbol{v}},q),({\boldsymbol{w}},r))\lesssim|||({\boldsymbol{v}},q)|||_{h}|||({\boldsymbol{w}},r)|||_{h}}$
(24)

PROOF. Define
$[{\boldsymbol{n}}\cdot{\boldsymbol{v}}]={\boldsymbol{n}}_{1}\cdot{\boldsymbol{v}}_{1}+{\boldsymbol{n}}_{2}\cdot{\boldsymbol{v}}_{2}$
for an interior face shared by elements $T_{1}$ and $T_{2}$ and
$[{\boldsymbol{n}}\cdot{\boldsymbol{v}}]={\boldsymbol{n}}\cdot{\boldsymbol{v}}$
for a face at the boundary belonging to element $T$. Noting that for a face
$F$ we then have
$[{\boldsymbol{n}}\cdot{\boldsymbol{v}}({\boldsymbol{x}}_{E})]=0$ for the
midpoint ${\boldsymbol{x}}_{E}$ of the edge $E$. Therefore
$([{\boldsymbol{n}}\cdot{\boldsymbol{v}}],1)_{F}=0$, since the quadrature
formula based on the midpoints of the edges on a triangle is exact for
quadratic polynomials. We therefore have
$([{\boldsymbol{n}}\cdot{\boldsymbol{v}}],q)_{F}=([{\boldsymbol{n}}\cdot{\boldsymbol{v}}],(I-P_{0,F})q)_{F}$,
where $P_{0,F}$ is the $L^{2}$ projection on constants on the face $F$. With
these preparations at hand we obtain the following bound

$\displaystyle\sum_{T\in\mathcal{T}_{h}}(\nabla\cdot{\boldsymbol{v}},q)_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{v}},q)_{\partial
T}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}(\nabla\cdot{\boldsymbol{v}},q)_{T}-\sum_{F\in\mathcal{F}_{h}}((I-P_{0,F})[{\boldsymbol{n}}\cdot{\boldsymbol{v}}],(I-P_{0,F})q)_{F}$
(25)
$\displaystyle\leq\sum_{T\in\mathcal{T}_{h}}\|\nabla\cdot{\boldsymbol{v}}\|_{T}\|q\|_{T}+\sum_{F\in\mathcal{F}_{h}}\|(I-P_{0,F})[{\boldsymbol{n}}\cdot{\boldsymbol{v}}]\|_{F}\|(I-P_{0,F})q\|_{F}$
(26)
$\displaystyle\leq\sum_{T\in\mathcal{T}_{h}}\|\nabla\cdot{\boldsymbol{v}}\|_{T}\|q\|_{T}+\sum_{F\in\mathcal{F}_{h}}h\|\nabla_{F}[{\boldsymbol{n}}\cdot{\boldsymbol{v}}]\|_{F}\|\nabla_{F}q\|_{F}$
(27)
$\displaystyle\leq\sum_{T\in\mathcal{T}_{h}}\|\nabla\cdot{\boldsymbol{v}}\|_{T}\|q\|_{T}+\|\nabla{\boldsymbol{v}}\|_{T}h\|\nabla
q\|_{T}$ (28)

which proves (23). Finally, (24) follows directly from (23) and the Cauchy-
Schwarz inequality.

###### Remark 3.3

For $\tilde{b}$ defined in (18) we directly have

$\displaystyle\tilde{b}({\boldsymbol{v}},q)\leq\|\nabla\cdot{\boldsymbol{v}}\|_{\Omega}\|q\|_{\Omega}$
(29)

since there are no trace terms on the boundary of the elements.

###  3.3 Interpolation

We shall now define interpolants for the finite element space. Starting with
the pressure space we let $\pi_{h,p}:L^{2}(\Omega)\rightarrow Q_{h}$ be a
Clement interpolant. We then have the standard estimate

$\|q-\pi_{h,p}q\|_{H^{m}(\Omega)}\lesssim h^{k-m}\|q\|_{H^{k}(\Omega)},\qquad
0\leq m\leq k\leq 2$ (30)

To construct an interpolant for the velocity space we use component-wise
Scott-Zhang interpolation to satisfy the Dirichlet boundary conditions (in the
nodes),

$\pi_{h,u}{\boldsymbol{v}}=[\pi_{h,SZ}v_{i}]_{i=1}^{3}$ (31)

Here we also have the interpolation estimate

$\|{\boldsymbol{v}}-\pi_{h,u}{\boldsymbol{v}}\|_{H^{m}(\Omega)}\lesssim
h^{k-m}\|{\boldsymbol{v}}\|_{H^{k}(\Omega)},\qquad 0\leq m\leq k\leq 2$ (32)

From here on we simplify the notation and write $\pi_{h,p}=\pi_{h}$ and
$\pi_{h,u}=\pi_{h}$ and interpret the operator in the correct way depending on
in which space the argument reside. Combining the estimates we get

$\displaystyle|||({\boldsymbol{v}},q)-(\pi_{h}{\boldsymbol{v}},\pi_{h}q)|||_{h}\lesssim
h(\|{\boldsymbol{v}}\|_{H^{2}(\Omega)}+\|q\|_{H^{1}(\Omega)})$ (33)

###  3.4 Stability Analysis

We first recall the following standard result from [1].

###### Theorem 3.1

If there is a constant such that

$\boxed{\|q\|_{\Omega}+h\|\nabla q\|_{\Omega}\lesssim\sup_{{\boldsymbol{v}}\in
W_{h}}\frac{b(q,{\boldsymbol{v}})}{\|{\boldsymbol{v}}\|_{a_{h}}}}$ (34)

Then there is a constant such that

$\boxed{|||({\boldsymbol{v}},q)|||_{h}\lesssim\sup_{({\boldsymbol{v}},q)\in
W_{h}\times
Q_{h}}\frac{A_{h}(({\boldsymbol{v}},q),({\boldsymbol{w}},r))}{|||({\boldsymbol{w}},r)|||_{h}}}$
(35)

We shall now prove that (34) holds for the nonconforming space $W_{h}\times
Q_{h}$ using an approach called Verfürth’s trick [8], which proceeds in two
steps.

######  Lemma 3.2 (Step 1)

There are constants $c_{1}$ and $c_{2}$ such that

$\boxed{\sup_{{\boldsymbol{v}}\in
W_{h}}\frac{b({\boldsymbol{v}},q)}{\|{\boldsymbol{v}}\|_{a_{h}}}\geq
c_{1}\|q\|-c_{2}h\|\nabla q\|}$ (36)

PROOF. For each $q\in Q_{h}$ there exists a ${\boldsymbol{v}}\in W$ such that

$\displaystyle
c_{1}\|q\|_{\Omega}\leq\frac{(\nabla\cdot{\boldsymbol{v}},q)_{\Omega}}{\|\nabla{\boldsymbol{v}}\|_{\Omega}}$
(37)

We shall now replace ${\boldsymbol{v}}\in W$ by the interpolant
$\pi_{h}{\boldsymbol{v}}$ and estimate the remainder term as follows

$\displaystyle-\frac{(\pi_{h}{\boldsymbol{v}},\nabla
q)_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}$
$\displaystyle=-\frac{({\boldsymbol{v}},\nabla
q)_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}+\frac{({\boldsymbol{v}}-\pi_{h}{\boldsymbol{v}},\nabla
q)_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}$ (38)
$\displaystyle=\frac{(\nabla\cdot{\boldsymbol{v}},\nabla
q)_{\Omega}}{\|\nabla{\boldsymbol{v}}\|_{\Omega}}\frac{\|\nabla{\boldsymbol{v}}\|_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}+\frac{({\boldsymbol{v}}-\pi_{h}{\boldsymbol{v}},\nabla
q)_{\Omega}}{\|\nabla{\boldsymbol{v}}\|_{\Omega}}\frac{\|\nabla{\boldsymbol{v}}\|_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}$
(39) $\displaystyle\geq
c_{1}\|q\|_{\Omega}-\frac{\|{\boldsymbol{v}}-\pi_{h}{\boldsymbol{v}}\|_{\Omega}\|\nabla
q\|_{\Omega}}{\|\nabla{\boldsymbol{v}}\|_{\Omega}}\frac{\|\nabla{\boldsymbol{v}}\|_{\Omega}}{\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}}$
(40) $\displaystyle\geq c_{1}\|q\|_{\Omega}-c_{2}h\|\nabla q\|_{\Omega}$ (41)

Here we used partial integration together with the continuity of $q$ and the
boundary condition ${\boldsymbol{v}}=0$ on $\partial\Omega$, the interpolation
estimate (32) and the boundedness
$\|\nabla\pi_{h}{\boldsymbol{v}}\|_{\Omega}\lesssim\|\nabla{\boldsymbol{v}}\|_{\Omega}$
of the interpolation operator.

######  Lemma 3.3 (Step 2)

Under the assumption (6) there exists $c_{3}>0$ such that

$\boxed{\sup_{{\boldsymbol{v}}\in
W_{h}}\frac{b({\boldsymbol{v}},q)}{\|{\boldsymbol{v}}\|_{a_{h}}}\geq
c_{3}h\,\|\nabla q\|}$ (42)

PROOF. Using partial integration we have the identity

$\displaystyle
b({\boldsymbol{v}},q)=\sum_{T\in\mathcal{T}_{h}}-({\boldsymbol{v}},\nabla
q)_{\mathcal{T}_{h}}$ (43)

Observing that $\nabla p$ is element-wise constant we may apply the quadrature
formula

$\int_{T}w({\boldsymbol{x}})dT\approx\int_{T}\sum_{E\in\mathcal{E}_{h}(T)}w({\boldsymbol{x}}_{E})\varphi_{T,E}({\boldsymbol{x}}_{E})dT=\frac{|T|}{6}\sum_{E\in\mathcal{E}_{h}(T)}w({\boldsymbol{x}}_{E})$
(44)

which is exact for $w\in V_{h}$, to obtain

$\displaystyle\sum_{T\in\mathcal{T}_{h}}-({\boldsymbol{v}},\nabla
q)_{\mathcal{T}_{h}}$
$\displaystyle=\frac{|T|}{6}\sum_{T\in\mathcal{T}_{h}}\sum_{E\in\mathcal{E}_{h}(T)}-{\boldsymbol{v}}({\boldsymbol{x}}_{E})\cdot\nabla
p({\boldsymbol{x}}_{E})$ (45)

where $\\{\varphi_{E}\\}_{E\in\mathcal{E}_{h}}$ is the global basis in
$V_{h}$, $\mathcal{E}_{h}(T)$ is the set of edges belong to element $T$, and
$\varphi_{T,E}=\varphi_{E}|_{T}$.

Since $q$ is continuous it follows that the tangent derivative
${\boldsymbol{t}}_{E}\cdot\nabla q$ along each edge $E$, with unit tangent
vector ${\boldsymbol{t}}_{E}$, is also continuous and thus taking

${\boldsymbol{v}}_{*}=\sum_{E\in\mathcal{E}_{h}^{I}}-h^{2}({\boldsymbol{t}}_{E}\cdot\nabla
q){\boldsymbol{t}}_{E}\varphi_{E}=\sum_{T\in\mathcal{T}_{h}}\sum_{E\in\mathcal{E}^{I}_{h}(T)}-h^{2}({\boldsymbol{t}}_{E}\cdot\nabla
q){\boldsymbol{t}}_{E}\varphi_{T,E}$ (46)

where $\mathcal{E}_{h}^{I}\subset\mathcal{E}_{h}$ is the set of interior
edges, we get

$\displaystyle\sum_{T\in\mathcal{T}_{h}}-(\nabla
q,{\boldsymbol{v}})_{\mathcal{T}_{h}}$
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}\frac{|T|}{6}\sum_{E\in\mathcal{E}_{h}^{I}(T)}h^{2}({\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E}))^{2}\gtrsim\sum_{T\in\mathcal{T}_{h}}h^{2}\|\nabla
q\|^{2}_{T}$ (47)

since $\nabla q$ is element-wise constant and there are, by assumption, three
linearly independent tangent vectors in the set
$\\{{\boldsymbol{t}}_{E}\\}_{E\in\mathcal{E}_{h}(T)}$ for each element
$T\in\mathcal{T}_{h}$. Finally, noting that

$\displaystyle\|\nabla v_{*}\|_{T}^{2}\lesssim
h^{4}\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E}))^{2}\|\nabla\varphi_{T,E}\|_{T}^{2}\lesssim
h^{4}\sum_{E\in\mathcal{E}^{I}_{h}(T)}|\nabla
q({\boldsymbol{x}}_{E})|^{2}h^{-2}h^{3}\lesssim h^{2}\|\nabla q\|_{T}^{2}$
(48)

where we used an inverse bound to conclude that
$\|\nabla\varphi_{E}\|_{T}^{2}\lesssim h^{-2}\|\varphi_{E}\|_{T}^{2}\lesssim
h^{-2}h^{3}$, and summing over $T$ gives

$\displaystyle\|v_{*}\|_{\Omega}\lesssim h\|\nabla q\|_{\Omega}$ (49)

Combining the estimates we get the desired result since

$\displaystyle\frac{b(v_{*},q)}{\|\nabla
v_{*}\|_{\Omega}}\gtrsim\frac{h^{2}\|\nabla q\|^{2}_{\Omega}}{\|\nabla
v_{*}\|_{\Omega}}\gtrsim h\|\nabla q\|_{\Omega}\frac{h\|\nabla
q\|_{\Omega}}{\|\nabla v_{*}\|_{\Omega}}\gtrsim h\|\nabla q\|_{\Omega}$ (50)

###### Remark 3.4

For the alternative form $\tilde{b}$ we get the more complicated expression

$\displaystyle\tilde{b}({\boldsymbol{v}},q)=\sum_{T\in\mathcal{T}_{h}}-({\boldsymbol{v}},\nabla
q)_{T}+({\boldsymbol{n}}\cdot{\boldsymbol{v}},q)_{\partial T}$ (51)

where the trace term on the boundary of $T$ does not vanish. We will however
show in A that with the same choice of ${\boldsymbol{v}}_{*}$ the trace term
can be shown to be dominated by the bulk term on each element. The proof is
based on mapping to the reference element and explicit computation of the two
integrals. Thus Lemma 3.3 also holds for the form $\tilde{b}$.

###### Lemma 3.4

There is a constant such that the inf-sup condition (34) holds.

PROOF. Multiply (36) by $c_{3}$ and (42) by $2c_{2}$ and add up to find

$\displaystyle(c_{3}+2c_{2})\sup_{{\boldsymbol{v}}\in
W_{h}}\frac{b(q,{\boldsymbol{v}})}{\|{\boldsymbol{v}}\|_{a_{h}}}\geq
c_{3}c_{1}\|q\|_{\Omega}-c_{3}c_{2}h\|\nabla
q\|_{\Omega}+2c_{3}c_{2}h\,\|\nabla
q\|_{\Omega}=c_{3}c_{1}\|q\|_{\Omega}+c_{3}c_{2}h\,\|\nabla q\|_{\Omega}$ (52)

Thus (34) holds with the hidden constant
$c_{3}=\min(c_{1},c_{2})/(c_{3}+2c_{2})$.

###  3.5 Error Analysis

###### Theorem 3.2

There is a constant such that

$\displaystyle\boxed{|||({\boldsymbol{u}}-{\boldsymbol{u}}_{h},p-p_{h})|||_{h}\lesssim
h\Big{(}\|{\boldsymbol{u}}\|_{H^{3}(\Omega)}+\|p\|_{H^{1}(\Omega)}\Big{)}}$
(53)

PROOF. We first split the error in an interpolation error part and a discrete
part

$\displaystyle|||({\boldsymbol{u}},p)-({\boldsymbol{u}}_{h},p_{h})|||_{h}\leq|||({\boldsymbol{u}},p)-(\pi_{h}{\boldsymbol{u}},\pi_{h}p)|||_{h}+|||(\pi_{h}{\boldsymbol{u}},\pi_{h}p)-({\boldsymbol{u}}_{,}p_{h})|||_{h}$
(54)

The first term can be estimated using the interpolation error estimate (33).
To estimate the discrete part of the error we employ the inf-sup condition
(35) to obtain

$\displaystyle|||(\pi_{h}{\boldsymbol{u}}-{\boldsymbol{u}}_{h},\pi_{h}p-p_{h})|||_{h}\lesssim\sup_{({\boldsymbol{v}},q)\in
W_{h}\times
Q_{h}}\frac{A_{h}((\pi_{h}{\boldsymbol{u}}-{\boldsymbol{u}}_{h},\pi_{h}p-p_{h},({\boldsymbol{v}},q))}{|||({\boldsymbol{v}},q)|||_{h}}$
(55) $\displaystyle\lesssim\sup_{({\boldsymbol{v}},q)\in W_{h}\times
Q_{h}}\frac{A_{h}((\pi_{h}{\boldsymbol{u}}-{\boldsymbol{u}},\pi_{h}p-p),({\boldsymbol{v}},q))}{|||({\boldsymbol{v}},q)|||_{h}}+\sup_{({\boldsymbol{v}},q)\in
W_{h}\times
Q_{h}}\frac{A_{h}({\boldsymbol{u}},p),({\boldsymbol{v}},q))-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}}{|||({\boldsymbol{v}},q)|||_{h}}$
(56)
$\displaystyle\lesssim|||(\pi_{h}{\boldsymbol{u}}-{\boldsymbol{u}},\pi_{h}p-p)|||_{h}+\sup_{({\boldsymbol{v}},q)\in
W_{h}\times
Q_{h}}\frac{A_{h}({\boldsymbol{u}},p),({\boldsymbol{v}},q))-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}}{|||({\boldsymbol{v}},q)|||_{h}}$
(57)

Here we used continuity (24) of the form $A_{h}$ for the first term which can
now be estimated using the interpolation error estimate (33). The second term
accounts for the consistency error and using partial integration we find that

$\displaystyle
A_{h}({\boldsymbol{u}},p),({\boldsymbol{v}},q))-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}=a_{h}({\boldsymbol{u}},{\boldsymbol{v}})-b({\boldsymbol{v}},p)+b({\boldsymbol{u}},q)-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}$
(58)
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}(\nabla{\boldsymbol{u}},\nabla{\boldsymbol{v}})_{T}+({\boldsymbol{v}},\nabla
p)_{T}-({\boldsymbol{u}},\nabla
q)_{T}-({\boldsymbol{f}},{\boldsymbol{v}})_{T}$ (59)
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}-(\Delta{\boldsymbol{u}},{\boldsymbol{v}})_{T}+(\nabla_{n}{\boldsymbol{u}},{\boldsymbol{v}})_{\partial
T}+({\boldsymbol{v}},\nabla
p)_{T}+(\nabla\cdot{\boldsymbol{u}},q)_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{u}},q)_{\partial
T}-({\boldsymbol{f}},{\boldsymbol{v}})_{T}$ (60)
$\displaystyle=\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,{\boldsymbol{v}}\otimes{\boldsymbol{n}})_{\partial
T}$ (61)
$\displaystyle=\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,[{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{F}$
(62)

where
$[{\boldsymbol{v}}\otimes{\boldsymbol{n}}]={\boldsymbol{v}}_{1}\otimes{\boldsymbol{n}}_{1}+{\boldsymbol{v}}_{2}\otimes{\boldsymbol{n}}_{2}$
for a face $F$ shared by elements $T_{1}$ and $T_{2}$ with
${\boldsymbol{v}}|_{T_{i}}={\boldsymbol{v}}_{i}$, $i=1,2.$ and
$[{\boldsymbol{v}}\otimes{\boldsymbol{n}}]={\boldsymbol{v}}\otimes{\boldsymbol{n}}$
for a face at the boundary. Then using the fact that
$[{\boldsymbol{v}}]={\boldsymbol{0}}$ in the midpoints of the edges and that
midpoint quadrature is exact for quadratic polynomials on a triangle it
follows that $(1,([{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{ij})_{F}=0$ and
therefore we may subtract the $L^{2}$-projection on constant functions on the
faces and then estimate the contributions using the following standard bounds

$\displaystyle\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,[{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{F}$
$\displaystyle=\sum_{F\in\mathcal{F}_{h}}((I-P_{0,F}){\boldsymbol{v}}\otimes\nabla,(I-P_{0,F})[{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{F}$
(63)
$\displaystyle\leq\sum_{F\in\mathcal{F}_{h}}\|(I-P_{0,F}){\boldsymbol{u}}\otimes\nabla\|_{F}\|(I-P_{0,F})[{\boldsymbol{v}}\otimes{\boldsymbol{n}}]\|_{F}$
(64)
$\displaystyle\lesssim\sum_{F\in\mathcal{F}_{h}}h\|\nabla_{F}({\boldsymbol{u}}\otimes\nabla)\|_{F}h\|\nabla_{F}[{\boldsymbol{v}}\otimes{\boldsymbol{n}}]\|_{F}$
(65)
$\displaystyle\lesssim\sum_{T\in\mathcal{T}_{h}}h\|{\boldsymbol{u}}\|_{H^{3}(T)}\|\nabla{\boldsymbol{v}}\|_{T}$
(66)

where $\nabla_{F}=(I-{\boldsymbol{n}}_{F}\otimes{\boldsymbol{n}})\nabla$, with
${\boldsymbol{n}}_{F}$ a unit normal to the $F$, is the tangential gradient to
the face $F$. Combing the bounds gives the desired estimate.

###### Remark 3.5

For the form $\tilde{b}$ we get a consistency error of the form

$\displaystyle
A_{h}({\boldsymbol{u}},p),({\boldsymbol{v}},q))-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}=a_{h}({\boldsymbol{u}},{\boldsymbol{v}})-b({\boldsymbol{v}},p)+b({\boldsymbol{u}},q)-({\boldsymbol{f}},{\boldsymbol{v}})_{\Omega}$
(67)
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}(\nabla{\boldsymbol{u}},\nabla{\boldsymbol{v}})_{T}-(\nabla\cdot{\boldsymbol{v}},p)_{T}+(\nabla\cdot{\boldsymbol{u}},q)_{T}-({\boldsymbol{f}},{\boldsymbol{v}})_{T}$
(68)
$\displaystyle=\sum_{T\in\mathcal{T}_{h}}-(\Delta{\boldsymbol{u}},{\boldsymbol{v}})_{T}+(\nabla_{n}{\boldsymbol{u}},{\boldsymbol{v}})_{\partial
T}+({\boldsymbol{v}},\nabla
p)_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{v}},p)_{\partial
T}-({\boldsymbol{f}},{\boldsymbol{v}})_{T}+(\nabla\cdot{\boldsymbol{u}},q)_{T}$
(69)
$\displaystyle=\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,{\boldsymbol{v}}\otimes{\boldsymbol{n}})_{\partial
T}-({\boldsymbol{n}}\cdot{\boldsymbol{v}},p)_{\partial T}$ (70)
$\displaystyle=\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,[{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{F}-([{\boldsymbol{n}}\cdot{\boldsymbol{v}}],p)_{\partial
T}$ (71)

which, using the same approach as in estimates (63-65), can be estimated by

$\displaystyle\sum_{F\in\mathcal{F}_{h}}({\boldsymbol{u}}\otimes\nabla,[{\boldsymbol{v}}\otimes{\boldsymbol{n}}])_{F}-([{\boldsymbol{n}}\cdot{\boldsymbol{v}}],p)_{\partial
T}$
$\displaystyle\lesssim\sum_{T\in\mathcal{T}_{h}}h\|{\boldsymbol{u}}\|_{H^{3}(T)}\|\nabla{\boldsymbol{v}}\|_{T}+h\|{\boldsymbol{v}}\|_{H^{1}(T)}\|p\|_{H^{1}(T)}$
(72)
$\displaystyle\lesssim\sum_{T\in\mathcal{T}_{h}}h(\|{\boldsymbol{u}}\|_{H^{3}(T)}+\|p\|_{H^{1}(T)})\|{\boldsymbol{v}}\|_{H^{1}(T)}$
(73)

##  4 NUMERICAL EXAMPLES

###  4.1 Convergence

We consider a problem in the ball with radius 1 and with center at the origin.
A fabricated solution is given by

${\boldsymbol{u}}=(x_{2}^{3}-x_{3}^{3},x_{1}^{3}-x_{3}^{3},-x_{1}^{3}-x_{2}^{3}),\quad
p=6(x_{1}x_{2}-x_{1}x_{3}-x_{2}x_{3})$ (74)

with ${\boldsymbol{f}}={\bf 0}$. The exact solution is used as Dirichlet data
and zero mean pressure is imposed by a Lagrange multiplier. We compare the
convergence of the pressure inconsistent method, using
$\tilde{b}({\boldsymbol{u}},q)$, to the pressure consistent method, using
$b({\boldsymbol{u}},q)$, in Fig. 3. The convergence is shown in $L_{2}$ for
the pressure and the velocity and in broken $H^{1}$ semi–norm for the
velocity. We note that the methods converge at the same rate, albeit with a
slightly larger error constant for the inconsistent method. The observed rates
from Fig. 3 are

$\displaystyle\|{\boldsymbol{u}}-{\boldsymbol{u}}_{h}\|_{L_{2}(\Omega)}={}$
$\displaystyle O(h^{2})$ (75)
$\displaystyle\|{\boldsymbol{u}}-{\boldsymbol{u}}_{h}\|_{a_{h}}={}$
$\displaystyle O(h)$ (76) $\displaystyle\|p-p_{h}\|_{L_{2}(\Omega)}\approx{}$
$\displaystyle O(h^{3/2})$ (77)

As we do not have full $P^{2}$ approximation for the velocity, the observed
rate of convergence of pressure is better than expected.

In Fig. 4 we compare the pressure solutions to the interpolated pressure on
the boundary, shown on the finest mesh in the sequence used for the
convergence study.

###  4.2 Laplacian Form vs. Strain Form

In this example we show that the strain formulation of Stokes equations poses
no problem as we have a Korn inequality for our approximation. We consider a
Poiseuille type problem in the domain $\Omega=(0,3)\times(0,1)\times(0,1/10)$
with boundary conditions ${\boldsymbol{u}}=(x_{2}(1-x_{2}),0,0)$ at $x_{1}=0$,
at $x_{2}=0$, and at $x_{2}=1$, and with $u_{3}=0$ at $x_{3}=0$ and at
$x_{3}=1/10$. In Fig. 6 we show the velocity field in the
$(x_{1},x_{2})$–plane and we note that the strain formulation gives a stress
free condition at the outflow. In Fig. 6 we show the corresponding pressure.
The computations were made using the $\tilde{b}$–form of the side condition.

Figure 1: The reference element and the enumeration of the six degrees of
freedom. Figure 2: The normal ${\boldsymbol{n}}$ and the three edge vectors
$\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2},\boldsymbol{E}_{3}\\}$. Figure 3:
Convergence of velocity and pressure for the pressure inconsistent
$\tilde{b}$–form and the consistent $b$–form.

Figure 4: Pressure solutions: inconsistent (left), consistent (center)
interpolated (right).

Figure 5: Velocity for the Laplacian form (left) and strain form (right).

Figure 6: Pressure for the Laplacian form (left) and strain form (right).

##  Appendix A INF-SUP CONDITION FOR THE $\boldmath{\tilde{b}}$ FORM

We shall now verify that the the inf-sup condition (34) holds also for the
form $\tilde{b}$. The element contribution for $\tilde{b}$ is

$\displaystyle\tilde{b}_{T}({\boldsymbol{v}},q)$
$\displaystyle=(\nabla\cdot{\boldsymbol{v}},q)_{T}=-({\boldsymbol{v}},\nabla
q)_{T}+({\boldsymbol{n}}\cdot{\boldsymbol{v}},q)_{\partial T}$ (78)

and as in the proof of Lemma 3.3 we take

${\boldsymbol{v}}_{*}=-\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E})){\boldsymbol{t}}_{E}\varphi_{E}$ (79)

which gives

$\displaystyle\tilde{b}_{T}({\boldsymbol{v}},q)$
$\displaystyle=\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E})\varphi_{E},{\boldsymbol{t}}_{E}\cdot\nabla
q)_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}({\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E}))\varphi_{E},q-P_{0,T}q)_{\partial T}$ (80)

where $P_{0,T}$ is the $L^{2}$-projection on constants on $T$. Here we used
the fact that

$({\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}{\boldsymbol{t}}_{E}\cdot\nabla
q({\boldsymbol{x}}_{E}))\varphi_{E},1)_{\partial T}=0$ (81)

to subtract $P_{0,T}q$. Identity (81) holds since if $F$ is a face that has
$E$ as one of its edges then ${\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}=0$ and
if $F$ does not have $E$ as an edge then $\varphi_{E}$ is zero in the
midpoints, which imply $(\varphi_{E},1)_{F}=0$ since midpoint quadrature is
exact for quadratic polynomials on triangles.

Next using the fact that $q$ is linear on $T$ we may write

$q=a+{\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})$ (82)

where ${\boldsymbol{x}}_{T}$ is the center of gravity of $T$. We then have
$(I-P_{0,T})q={\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})$
and

$\displaystyle\tilde{b}_{T}({\boldsymbol{v}}_{*},q)$
$\displaystyle=\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}\varphi_{E}({\boldsymbol{x}}),{\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})_{T}-({\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})\varphi_{E}({\boldsymbol{x}}),{\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T}))_{\partial
T}$ (83)

We now study the contributions for each $E\in\mathcal{E}_{h}^{I}$. The bulk
term may be directly computed using (44) as

$({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}\varphi_{E}({\boldsymbol{x}}),{\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})_{T}=({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}\int_{T}\varphi_{E}({\boldsymbol{x}})=({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}|T|/6$
(84)

The formula holds for general elements since the Jacobian of the affine
mapping is constant. It remains to study the boundary contribution

$\displaystyle
g({\boldsymbol{b}})=-{\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}\int_{\partial
T}{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}\varphi_{E}({\boldsymbol{x}}){\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})$
(85)

which is a quadratic function in ${\boldsymbol{b}}$.

#### Computation on the Reference Element

Let us first look at the reference element $\hat{T}$ as defined in Section 2.
Expanding ${\boldsymbol{b}}$ in the orthonormal basis
$\\{{\boldsymbol{e}}_{1},{\boldsymbol{t}}_{E},\boldsymbol{s}_{E}\\}$, we
obtain by symmetry

$\displaystyle\int_{\partial
T}{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}\varphi_{E}{\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})=\begin{cases}0&{\boldsymbol{b}}={\boldsymbol{e}}_{1}\\\
0&{\boldsymbol{b}}=\boldsymbol{s}_{E}\end{cases}$ (86)

Therefore,

$\displaystyle
g({\boldsymbol{b}})=-({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}\int_{\partial
T}{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}\varphi_{E}({\boldsymbol{x}}){\boldsymbol{t}}_{E}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})=-2({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}\underbrace{\int_{F^{*}(E)}{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}\varphi_{E}({\boldsymbol{x}}){\boldsymbol{t}}_{E}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})}_{I}$
(87)

where $F^{*}(E)$ is one of the two faces faces to which $E$ does not belong
and we again used symmetry to conclude that the contributions from the two
faces are identical. We thus obtain

$b({\boldsymbol{v}}_{*},q)=\sum_{E\in\mathcal{E}_{h}(T)}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}(|T|/6-2I)$
(88)

Here $|T|/6=(8/3)/6=4/9$ and direct computation gives $2I=4/15$ and thus

$\displaystyle\boxed{b({\boldsymbol{v}}_{*},q)=\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}\left(\frac{4}{9}-\frac{4}{15}\right)=\frac{8}{45}\sum_{E\in\mathcal{E}_{h}^{I}(T)}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}}$
(89)

#### Mapped Element

Next consider a mapped element

$\displaystyle F:{\hat{T}}\rightarrow T$ (90)

where $F$ is an affine map of the form

${\boldsymbol{x}}=F(\hat{\boldsymbol{x}})=A\hat{\boldsymbol{x}}+{\boldsymbol{x}}_{T}$
(91)

which means that

${\boldsymbol{x}}-{\boldsymbol{x}}_{T}=F(\hat{\boldsymbol{x}})=A\hat{\boldsymbol{x}}$
(92)

where we use the usual hat notation for quantities on the reference element.
Tangent vectors are mapped to tangent vectors

$A\hat{\boldsymbol{t}}_{E}=\|A\hat{\boldsymbol{t}}_{E}\|{\boldsymbol{t}}_{E}$
(93)

where

$\|A\hat{\boldsymbol{t}}_{E}\|=h_{E}/h_{\hat{E}}$ (94)

and given ${\boldsymbol{b}}$ associated with $T$ we define

$\hat{\boldsymbol{b}}_{E}=A^{T}{\boldsymbol{b}}/\|A\hat{{\boldsymbol{t}}}_{E}\|$
(95)

which leads to the identity

$\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}_{E}={\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}$
(96)

As above the element contribution takes the form

$\displaystyle\int_{T}{\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}\varphi_{E}({\boldsymbol{x}}){\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}$
$\displaystyle=({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}|T|/6=|T|\,|\hat{T}|^{-1}(\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}_{E})^{2}|\hat{T}|/6$
(97)

Next for the boundary contribution

$\displaystyle{\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}}\int_{F}{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}\varphi_{E}({\boldsymbol{x}}){\boldsymbol{b}}\cdot({\boldsymbol{x}}-{\boldsymbol{x}}_{T})dF$
(98)
$\displaystyle\qquad=\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}\int_{\hat{F}}\hat{\varphi}_{\hat{E}}(\hat{\boldsymbol{x}})(A^{T}{\boldsymbol{b}})\cdot\hat{{\boldsymbol{x}}}\,{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}|A|_{F}d\hat{F}$
(99)
$\displaystyle\qquad=|T|\,|\hat{T}|^{-1}\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}\int_{\hat{F}}\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{t}}_{E}\,\hat{\varphi}_{\hat{E}}(\hat{\boldsymbol{x}})\,\|A\hat{\boldsymbol{t}}_{\hat{E}}\|^{-1}A^{T}{\boldsymbol{b}}\cdot\hat{{\boldsymbol{x}}}d\hat{F}$
(100)
$\displaystyle\qquad=|T|\,|\hat{T}|^{-1}\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}\int_{\hat{F}}\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{t}}_{E}\,\hat{\varphi}_{\hat{E}}(\hat{\boldsymbol{x}})\,\hat{\boldsymbol{b}}\cdot\hat{{\boldsymbol{x}}}d\hat{F}$
(101)

Here $|A|_{F}$ is the Jacobian associated with the mapping
$A:\hat{F}\rightarrow F$ and we used the identity

$\displaystyle{\boldsymbol{n}}\cdot{\boldsymbol{t}}_{E}|A|_{F}=|T|\,|\hat{T}|^{-1}\|A\hat{\boldsymbol{t}}_{\hat{E}}\|^{-1}\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{t}}_{\hat{E}}$
(102)

which we verify below. We conclude that after transformation back to the
reference element we get an expression that has the same form as in the
reference element case and thus we may apply (89) to get

$\displaystyle
b_{T}({\boldsymbol{v}}_{*},q)=|T|\,|\hat{T}|^{-1}\frac{8}{45}\sum_{E\in\mathcal{E}_{h}^{I}}(\hat{\boldsymbol{t}}_{E}\cdot\hat{\boldsymbol{b}}_{E})^{2}=|T|\,|\hat{T}|^{-1}\frac{8}{45}\sum_{E\in\mathcal{E}_{h}^{I}}({\boldsymbol{t}}_{E}\cdot{\boldsymbol{b}})^{2}=|\hat{T}|^{-1}\frac{8}{45}\sum_{E\in\mathcal{E}_{h}^{I}}\|{\boldsymbol{t}}_{E}\cdot\nabla
q\|_{T}^{2}$ (103)

Summing over $T\in\mathcal{T}_{h}$ gives

$\displaystyle\boxed{b({\boldsymbol{v}}_{*},q)=\sum_{T\in\mathcal{T}_{h}}b_{T}({\boldsymbol{v}}_{*},q)=|\hat{T}|^{-1}\frac{8}{45}\sum_{T\in\mathcal{T}_{h}}\sum_{E\in\mathcal{E}_{h}^{I}}\|{\boldsymbol{t}}_{E}\cdot\nabla
q\|_{T}^{2}\gtrsim\sum_{T\in\mathcal{T}_{h}}\|\nabla q\|^{2}_{T}}$ (104)

and we note that the deformation of the element only has an effect on the last
inequality.

#### Verification of (102)

Let $\\{\boldsymbol{E}_{1},\boldsymbol{E}_{2},\boldsymbol{E}_{3}\\}$ be edge
vectors to $T$ such that, by well known relations with
$\det({\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}})$ as the determinant
of the matrix whose columns are ${\boldsymbol{a}}$, ${\boldsymbol{b}}$, and
${\boldsymbol{c}}$,

$6|T|=\det(\boldsymbol{E}_{1},\boldsymbol{E}_{2},\boldsymbol{E}_{3})$ (105)

and

$2|F|=\det({\boldsymbol{n}},\boldsymbol{E}_{2},\boldsymbol{E}_{1})$ (106)

see Fig. 2. We then have

$\displaystyle 6|T|$
$\displaystyle=\det(\boldsymbol{E}_{1},\boldsymbol{E}_{2},\boldsymbol{E}_{3})=\det(\boldsymbol{E}_{1},\boldsymbol{E}_{2},(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}}){\boldsymbol{n}})=\det(\boldsymbol{E}_{1},\boldsymbol{E}_{2},(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}}){\boldsymbol{n}})$
(107)
$\displaystyle\qquad=(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}})\det(\boldsymbol{E}_{1},\boldsymbol{E}_{2},{\boldsymbol{n}})=-(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}})\det({\boldsymbol{n}},\boldsymbol{E}_{2},\boldsymbol{E}_{1})=-(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}})2|F|$
(108)

Therefore

$\displaystyle-(\boldsymbol{E}_{3}\cdot{\boldsymbol{n}})|F|=3|T|=|T|\,|{\hat{T}}|^{-1}3|{\hat{T}}|=-|T|\,|{\hat{T}}|^{-1}(\hat{\boldsymbol{E}}_{3}\cdot\hat{\boldsymbol{n}})|\hat{F}|$
(109)

or

$\displaystyle
h_{E}({\boldsymbol{t}}_{3}\cdot{\boldsymbol{n}})|F|=|T|\,|{\hat{T}}|^{-1}\hat{h}_{E}(\hat{\boldsymbol{t}}_{3}\cdot\hat{\boldsymbol{n}})|\hat{F}|$
(110)

which finally imply

$\displaystyle({\boldsymbol{t}}_{3}\cdot{\boldsymbol{n}})|F|=|T|\,|{\hat{T}}|^{-1}\hat{h}_{E}h_{E}^{-1}(\hat{\boldsymbol{t}}_{3}\cdot\hat{\boldsymbol{n}})|\hat{F}|=|T|\,|{\hat{T}}|^{-1}\|A\hat{{\boldsymbol{t}}}_{3}\|^{-1}(\hat{\boldsymbol{t}}_{3}\cdot\hat{\boldsymbol{n}})|\hat{F}|$
(111)

#### Acknowledgement

This research was supported in part by the Swedish Research Council Grants
Nos. 2017-03911, 2018-05262, 2021-04925, and the Swedish Research Programme
Essence.

## References

  * [1] F. Brezzi and M. Fortin.  Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics.  Springer-Verlag, New York, 1991.
  * [2] M. Crouzeix and P.-A. Raviart.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I.  Revue Française Automatique Informatique Recherche Opérationnelle Série Rouge, 7(R-3):33–75, 1973.
  * [3] P. Hansbo.  A nonconforming rotated $Q_{1}$ approximation on tetrahedra.  Computer Methods in Applied Mechanics and Engineering, 200(9-12):1311–1316, 2011.
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Najmeh Mirian<EMAIL_ADDRESS>
#  Generation and Measurement of Ultrashort Free Electron Laser Pulses in
Ultraviolet and Soft-X-ray Spectral Regions

N. S. Mirian  formerly at Elettra-Sincrotrone Trieste S.C.p.A., 34149 Trieste,
Italy

###### Abstract

Over the last few years, tremendous progress has been gained in the generation
and application of ultrashort radiation pulses. Recently, free-electron lasers
generating ultrashort pulses with high peak power from the extreme ultraviolet
(EUV) to the soft-X-ray region are opening a wide range of new scientific
opportunities. Taking advantage of this short timescale permits probing
ultrafast, out-of-equilibrium dynamics and the high intensities are key for
nonlinear optics. The core structure of the extremely important light elements
carbon, nitrogen, and oxygen can be accessed by soft-X-ray wavelengths by
providing chemical sensitivity. Externally seeded free-electron lasers
generate coherent pulses with the ability to be synchronized with femtosecond
accuracy. In this contribution, we present new achievements in the generation
of coherent ultrashort pulses in the range of EUV to the soft X-ray in
externally seeded FELs. In particular, we present the recently successful
robust experiment at FERMI in Trieste, where few-femtosecond extreme-
ultraviolet pulses were generated and characterized in terms of energy, and
duration via autocorrelation.

###### keywords:

Attosecond pulse, femtosecond pule, free electron laser, ultrashort pulse,
seeded FEL

##  1 INTRODUCTION

The study of the temporal evolution of matter down to the sub-femtosecond time
scale and atomic space scale is essential for chemical, physical and
biological processes. In this respect, free-electron lasers (FELs) open a new
door to a wide range of new scientific opportunities by providing
unprecedented pulse properties in the soft and hard-X-ray spectral ranges, in
terms of brilliance, tuneability, intensity, coherence, and last but not
least, pulse duration. Intense ultrashort FEL pulses have paved the way for
time resolution and for the observation of nonlinear and ultrafast processes
in the study of the interaction of radiation with a matter involving shallow
and deep-core electron levels.

In general, FEL sources can be operated in several different ways. To date,
most of the existing short-wavelength FELs such as FLASH [1], LCLS [2], and
EuXFEL [3] are operated in the self-amplification of spontaneous emission
(SASE) [4]. Though SASE FELs are able to generate extremely high brilliance,
the temporal structure of the pulses includes a series of micro-pulses that
individually have random phases and highly fluctuating peak intensity and time
duration. Synchronization of SASE devices to external sources is normally
limited by the temporal jitter of the electron accelerator. Ultrashort pulses
in FELs can be generated in SASE mode. In SASE FELs the most straightforward
method consists in shortening the pulse region that has the right properties
to lase [5]. Several other schemes were proposed to reduce the pulse duration
at the femtosecond level or shorter. [6, 7, 8, 9, 10, 11, 12]

An alternative operation mode for an FEL is based upon ‘seeding’ techniques.
When the FEL operation wavelength is in the VUV to soft X-ray spectral range,
the FEL amplification can be initiated by a coherent radiation pulse generated
by an external laser, the seed. Seeded FELs produce output pulses with a well-
defined temporal shape and intensity stability. The availability of a seed of
sufficient intensity becomes increasingly critical as the wavelength
decreases. In the hard X-rays, the only seed available with sufficient
intensity consists of the radiation emitted by the beam itself, in a self-
seeded scheme [13] In externally seeded FELs, before emitting coherent
radiation, the electrons interact with an external coherent source (the seed,
generally at UV wavelengths) in a modulator, and then under given conditions,
one of the higher order harmonics of the seed is selected, and amplified in a
long amplifier, named radiator.

Two well-known seeded FEL schemes are the high-gain harmonic generation scheme
(HGHG) [14, 15] and the echo enable harmonic generation (EEHG).[16, 17] Fig. 1
shows the HGHG scheme. The electron beam after being accelerated in a linear
accelerator interacts with an external coherent laser in UV wavelengths at a
modulator (M1). The laser produces a sinusoidal energy modulation in the
electron beam. The beam then goes through the longitudinally dispersive
section (DS1). The dispersive section creates high harmonic bunching at short
wavelengths in the current distribution. The beam then enters the radiator (R)
and emits coherent light and exponentially amplifies it. HGHG scheme is able
to produce bunching in the electron beam down to the XUV region.

Figure 1: HGHG scheme

Fig. 2 presents the EEHG scheme. EEHG employs two setups of external laser and
dispersive section to efficiently generate bunching in the electron beam down
to soft X-rays. The first laser produces a sinusoidal energy modulation in the
electron beam at the first modulator (M1). Then the beam travels through a
strong longitudinally dispersive section (DS1) that folds over the sinusoids
and produces filaments in the beam phase space distribution. The second laser
then modulates the filaments’ distribution at the second modulator (M2).
Finally the second dispersive section (DS2) generates high harmonic bunching
at short wavelengths in the current distribution [18, 17, 16].

Figure 2: EEHG scheme

FERMI [15] is an externally seeded FEL user facility, based on the high-gain
harmonic generation scheme (HGHG) [14]. FERMI has two FEL beamlines, FEL-1 is
single-stage HGHG [19]and FEL-2 is based on two-stage HGHG, separated by
magnetic delay line to facilitate the fresh bunch technique [20]. Regarding
the generation and measurement of an ultrashort pulse in a short-wavelength
range at seeded FEL, several methods based on manipulation of the electron
beam, altering seed laser parameters, using the FEL amplification concept,
etc, have been proposed and demonstrated by the FERMI team in recent years. In
this contribution, we summarize the recent FERMI team activities with respect
to the first extended study of ultrashort seeded FEL pulse generation and of
the seeded FEL pulse temporal properties characterization.

Figure 3: The FEL rms pulse duration versus harmonic number. Blue squares
represent two-color gas photo-emission cross-correlation method results, green
triangles show high-resolution single-shot solid-state cross-correlation
method results. The Green dashed line shows the function $1/\sqrt{n}$, where n
is the harmonic number. The blue dotted line shows the optimized peak bunching
factor estimated by theory, scaling as about $7/(6n^{1/3})$. The red dots are
larger duration pulses obtained with an increased seed power or dispersion.
(see Fig.11 in Ref. 21)

##  2 Measurement and estimation of seeded FEL pulse

Characterization of the temporal intensity profile of free-electron laser
(FEL) pulses and determination of the pulse duration are extremely crucial for
exploring the new perspectives suggested by FEL sources. The analysis of
experimental data, particularly for nonlinear interactions requires accurate
knowledge of pulse peak power in the time domain. In the last decade, several
direct and indirect methods have been developed to illustrate reliable
temporal profiles, both on average and on a single-shot basis. S. Düsterer and
his colleagues in DESY utilized and compared nine different pulse duration
measurement methods to characterize the photon pulse duration, generated in
the wavelength range of 13.5 to 27 nm at FLASH, the self-amplified spontaneous
emission FEL at DESY in Hamburg [22, 23]. They represented autocorrelation
with gas phase and semiconductors detectors, and cross-correlation with
terahertz laser (for comparison see Fig.12 in Ref. 22).

FERMI team has measured the pulse shape of an extreme ultraviolet and soft
X-ray externally seeded FEL operating in high-gain harmonic generation mode by
different methods [21, 24, 25]. They characterized a single-shot
spectrotemporal of the seeded FEL pulses in a specific double-pulse
configuration by means of the spectral phase interferometry for direct
electric field reconstruction (SPIDER) technique[25]. This method is able to
show the reconstruction of the amplitude and phase of the pulses. However,
SPIDER method is applicable to a specific machine setup since it imposes
several constraints on the FEL operation.

In 2017, the FERMI team presented two different cross-correlation methods:
two-color photo-emission cross-correlation (method A, Sec. III A in Ref. 21)
and high-resolution single-shot solid-state cross-correlation (method B, Sec.
III B in Ref. 21). Basically in both cases, the FERMI team measured the
similarity of the unknown FEL pulse with an external optical laser. In that
work, the FEL temporal pulse measurement was carried out in a range of FEL
wavelengths, as different harmonics of the seed laser and machine settings,
and compared to the predictions of a theoretical model. Blue squares in Fig.
3) refer to measured rms pulse duration of different harmonic FEL
amplification by the two-color gas photo-emission cross-correlation method and
green triangles indicate the measurement results by high-resolution single-
shot solid-state cross-correlation method. The results obtained with the two
methods were in agreement with the proposed theory. In the theory, presented
in that paper, the pulse duration in seeded FEL performance is estimated by
the bunching factor parameter, the Fourier coefficient of the electron density
modulation at the resonant frequency of amplification. The amplitude of the
bunching factor in the time domain estimates the rms FEL pulse duration under
the conditions of low seed power and low dispersion, which should be around
$\sigma_{s}/\sqrt{n},$ where n is the number of seed harmonics and
$\sigma_{s}$ is the rms value of the seed laser. This function is shown by the
Green dashed line in Fig. 3. As well, it was proved that in the case of
optimal seed power and dispersion, the rms value of the FEL pulse duration is
approximated by the function $7\sigma_{s}/(6n^{1/3})$, presented by the dashed
blue line in Fig. 3.

##  3 femtosecond shaping of seeded FEL pulse by chirped seed

Though ultrafast pulse shaping techniques are well demonstrated in the
infrared and visible, direct application of such methods to an X-ray FEL is
not straightforward. In 2016, K. C. Prince et al. showed the necessity of
controlling and shaping the FEL spectral and temporal in Ref. 26. The ability
to control and shape the spectral and temporal of extreme-ultraviolet and soft
X-ray pulses produced by seeded FEL at the FERMI facility, has been
demonstrated by employing the precise manipulation of the linear frequency
chirp and intensity of the seed laser pulse as well the variation of
longitudinal dispersion. [27, 28] In this section, we discuss the experiments
in HGHG and EEHG performances, showing femtosecond shaping of seeded FEL
pulse.

###  3.1 HGHG

In general, the length of seeded FEL pulses is determined by the length of the
seed laser and is typically in the range between a few tens of fs and 100 fs.
Due to several mechanisms in the seeding process, shorter pulses (in the range
of a few fs) are not directly accessible with HGHG. However, fs time
resolution experiments can still be done taking advantage of the high degree
of longitudinal coherence of the produced pulses. Indeed, if the length of the
pulse is limited to tens of fs the phase along the pulse can be controlled
with much higher accuracy.

Exploiting the perfect phase locking between two harmonics of the same seed
laser makes it possible to perform coherent control experiments with
attosecond accuracy [26]. A proper combination of 3 and 4 harmonics of the
same seed was used for the generation of an attosecond pulse train in the UV,
presented by P.K. Maroju and the FERMI team in Ref. 29. All these methods for
accessing the attosecond time scale with tens of fs pulses rely on the
accurate control of the phase which is possible thanks to the presence of an
optical laser as the seed starting the emission process.

Figure 4: Normalized FEL pulse profile, simulated from bunching factor
function versus longitudinal dispersion $B=R_{56}$. The FWHM seed pulse
duration was assumed 50 fs and the FEL wavelength is the 7th harmonic of the
seed laser (seed wavelength is 260 nm ).

Dedicated experiments at FERMI showed in Ref. 27 that proper control of the
seed laser and electron beam parameters allows controlling the phase profile
in the FEL pulse. Thanks to the overbunching that can be created with a proper
adjustment of the seed laser intensity and longitudinal dispersion ($R_{56}$),
it is possible to induce a splitting of the FEL pulse. This allows generating
of two (or eventually more pulses) in the time domain. Suche a structure is
also visible in the spectral domain. This fact is presented in Fig. 4. We
estimated the shape of the FEL pulse bunching factor profile in the time
domain. Fig. 4 shows the map of the FEL pulse profile as a function of
longitudinal dispersion of the first chicane in the HGHG scheme (Ds1 in Fig.
1). At a certain point of dispersion, the FEL pulse splits into two or more
femtosecond pulses. The time separation of the pulses can be controlled by the
chirp and intensity parameters of the seed laser pulse. More information about
this technique is reported in Refs. 27, 30. In addition, if two phase-locked
pulses are used for seeding instead, it has been shown that a phase-locked FEL
pulse can be generated [31]. This again allows accessing the sub fs time
resolutions simply controlling the phase between the two seed laser pulses.

###  3.2 EEHG

Echo enable harmonic generation FEL performance was executed at FERMI in 2018
for half a year. FERMI demonstrated the first high-gain lasing of an EEHG FEL
in the soft X-ray region at 7.3 nm and 5.9 nm, which was the thirty-sixth and
forty-fifth harmonics of the seed wavelength [16]. The FERMI team prepared the
EEHG setup by altering the FEL-2 line which applies the double-stage HGHG
approach to extend the harmonic range up to 60 -70 [20]. In following the EEHG
demonstration, N. S. Mirian and FERMI team in Ref. 28 illustrated that
spectral and temporal control over the FEL pulse can be achieved by tuning the
second seed laser parameters. In the EEHG setup, the second seed laser
directly corresponds to the intensity modulations in the temporal domain.
Increasing second seed power develops the FEL spectrum intensity modulations.
Fig. 5-left shows the spectral maps (spectrum vs second seed laser energy) for
different values of the group delay dispersion (GDD) of the second seed laser;
the insets are theoretical spectral maps obtained from EEHG equations (see Eq.
(2) in Ref. 28 ) and agree relatively well with the experiment. Fig. 5-right
represents the calculated temporal domain reconstruction and phase of the FEL
pulses for the case of the spectrum indicated by the vertical dashed line in
the inset figures. In Fig. 5(a), we plot the results for a relatively high
(absolute) value of the GDD $\sim$ −4100 $fs^{2}$. Starting from the optimum
energy modulation that maximizes bunching, increasing the seed intensity, the
spectrum initially characterized by a single peak broadens until it develops
intensity modulations as a function of the wavelength due to the process of
electron overbunching and rebunching [27]. For such high GDD, this intensity
modulation directly corresponds to the intensity modulation in the temporal
domain due to the spectrotemporal equivalence. A split spectrum in the
wavelength domain indicated by the vertical dashed line in the inset of Fig.
5(a)-left, thus corresponds to few-femtosecond double-pulse structure with an
overall negative quadratic chirp in the temporal domain, as shown in Fig.
5(a)-right. The temporal separation between the pulses is controlled by the
GDD value. The spectral map for zero GDD, Fig. 5(b)-left, corresponds to a
Fourier transform limited FEL pulse structure with a flat phase within the
individual spikes, as demonstrated in Fig. 5(b)-right for the case of the
spectrum indicated by the vertical dashed line in the inset of Fig. 5(b)-left.
The spectral map for the case of a positive GDD of 2000 $fs^{2}$, is
represented in Fig. 5(c)-left and reconstruction of the phase and temporal
profile of the FEL pulse for the case of the spectrum indicated by the
vertical dashed line in the inset, are shown in Fig. 5(c)-right. This result
demonstrates deterministic control of the FEL phase in time and in the
spectrum domain by changing the parameters of second seed laser in EEHG setup.

Figure 5: Left: Evolution of the spectral shape as a function of second seed
laser energy with three different circumstances of the linear frequency chirp
on it. (a): $(GDD=-3500\,fs^{2})$ (b): $(GDD=0\,fs^{2}),$
(c):$(GDD=+2300\,fs^{2})$ . Insets are theoretical and simulated spectral maps
reproduced in the spectral domain. In all cases, the e-beam energy, energy
spread, current, and emittance were $\simeq$1.3 GeV, $\simeq$270 keV, 700 A,
and $\simeq$1 mm mrad, respectively. For (a) and (b), the e-beam (positive)
quadratic energy curvature was $\simeq$8 MeV/ps2. The maps in (a) and (b)
consist of 20 accumulated single-shot spectra at harmonic 37 for each value of
second seed energy, and data in (c) were taken 50 shots pf spectra at harmonic
36. First seed energy is $52\,\mu j$ ($A_{1}=4$ in simulation), $B_{1}=11.4$
and $B_{2}$ was in optimal value. Right: Time-domain normalized pulse shape
(blue solid line) and phase (dashed red line) of the time profile of the
bunching factor, obtained from the Fourier transform of the (calculated)
bunching spectra indicated along the dashed lines in the insets of right
figures for (a) $GDD=-3500\>fs^{2}$, (b) $GDD=0\,fs^{2}$, (c)
$GDD=2300\>fs^{2}$.

##  4 Chirped pulse amplification in free-electron laser

Chirped pulse amplification (CPA) in optical lasers is known as a
revolutionary technique. CPA approach allows the amplification of the
ultrashort laser pulse up to the petawatt level, by stretching out the laser
pulse temporally and spectrally first, then amplifying, and compressing the
pulse at the end. The stretching and compression applied by the devices with
the concept that the different color components of the pulse travel different
distances. The first femtosecond FEL by CPA was proposed by L.H. Yu et al. in
1994 [32]. Like the classical lasers, the CPA technique at seeded FEL relies
on stretching the seed pulse in time by means of a linear frequency chirp
before amplification, that enables us to extract energy from the whole
electron bunch and enhances the FEL pulse energy at saturation. To sustain the
FEL resonant condition, the large bandwidth of the seed laser is matched by
proper chirp in the energy of the electron beam.

D. Gauthier with the FERMI team in Ref. 33 presented the experimental
implementation of chirped pulse amplification on HGHG performance in the
extreme ultraviolet wavelength range. They imposed a positive linear frequency
chirp on the seed pulse and stretched it up to 290 fs. The chirped seed
manipulated the chirped electron beam in the modulator (M1 in Fig. 1). Then,
harmonic 7 of the seed was chosen for amplification in the radiator (R in Fig.
1). The final FEL pulses before and after compression were characterized
during the experiment. The temporal characterization was done by employing the
cross-correlation scheme, based on the ionization of a Helium gas sample by
the FEL pulse in the field of an intense infrared laser pulse with 90 fs FWHM
duration. The technical detail of measurement and compression of the FEL pulse
was reported in Ref. 33. Fig.3 of that reference summarized the results of the
experiment. The average FWHM spectral width was $4.46\times 10^{-2}$ nm, while
fluctuations between consecutive measurements were in order of few percent.
Three independently measured cross-correlation curves, associated with the
second sideband of the photo-electron spectrum, were presented for the non-
compressed case and for the maximum compression cases. After reconstruction of
the FEL pulses from cross-correlation measurements, the obtained FWHM pulse
duration for the case of no-compression was $\sim$143 fs (with fluctuations of
few percent). And after compression, the FWHM pulse duration was 40 fs which
was a significant shortening of the FEL pulse compared with the no-CPA case.
It is theoretically possible to implement this technique into the soft x-ray
spectral window with a pulse duration much below femtosecond, however, the
technique suffers from the limited transmission of the grating system required
to compress the pulses that at short wavelength can be lower than 1 percent.

The CPA performance result showed that CPA cannot only compensate for the
linear frequency chirp induced by seed control in the FEL pulse, but also
unwanted residuals generated by other sources, such as the quadratic curvature
of the electron-beam energy profile, seed transport, and so on.

Figure 6:  Up: the cascade superradiant scheme. The FEL-2 line was divided
into three sections. The electron beam was modulated with the coherent seed at
264.4 nm wavelength in modulator, M1. With the help of the longitudinal
dispersion of the first chicane, we selected the third harmonic and amplified
it in two undulator sections. In the next two undulator sections, we tuned the
resonance on the 6th harmonics of the seed laser (44 nm), and then in the last
radiator (R2) included six undulator sections, the harmonic 18th was
amplified. The longitudinal dispersion of the Delay line (DL) and second
dispersion section was set to zero. Down: Simulated radiation pulse (blue
solid line) and e-beam energy spread profile (red dashed line) after (a)
first, (b) second, and (c) third stage. Simulated spectrum profile (green
solid line) at (d) first, (e) second, and (f) third stage. Measured spectrum
after (g) first, (h) second, and (l) third stage.

##  5  superradiant spike in free-electron laser amplifier

Nonlinear FEL evolution and pulse propagation effects are important and
interesting methods to shorten the FEL pulse. Bonifacio and his collaborators
[34, 35] elaborated a regime where FEL evolution is dominated by the nonlinear
effects associated with saturation and by pulse propagation effects associated
with slippage. This regime is characterized by the propagation of a solitary
wave-like self-similar pulse with peak power scaling as $N^{2}$, where N is
the number of electrons[36]. Amplification of a soliton, superradiant is known
as a popular and robust method to generate ultrashort intense FEL pulse. The
interesting advantage of the superradiant approach with respect to the other
methods is that it can exploit the FEL dynamic process to beat the gain
bandwidth limit. FEL enters this regime when a short pulse grows up to
saturation while it is advancing on the electron that is delayed by the
wiggling motion in the undulator. The pulse peak power continues to grow
proportionally to the square of the distance covered along the undulator and
the pulse duration is proportional to the inverse of the square root of the
distance.
The propagation of the superradiant pulse is performed by the amplification of
harmonic emission and enhancement of the harmonic bunching on the head of the
pulse. This harmonic bunching can be used in a second radiator section, tuned
at one harmonic of the seed. This approach enables us to produce a
superradiant pulse at shorter wavelengths [37]. The pulse shortening via
superradiance in a single stage was demonstrated in the infrared (800 nm)
[38]. The possibility of having superradiant propagation in a two-stage
cascade was verified at SPARC [39]. In that case, the two stages were tuned at
400 nm at 200 nm respectively.
In 2020, N. S. Mirian and FERMI team demonstrated the generation of few-
femtosecond extreme-ultraviolet pulses via a multi-stage superradiant approach
and measured the pulse by autocorrelation method assisted by photo-emission of
Argon gas [40]. The FERMI team achieved pulses four times shorter, with higher
peak power, than in the standard high-gain harmonic generation mode and proved
that the pulse duration matched the Fourier transform limit of the spectral
intensity distribution.
The experiment was accomplished at the FERMI FEL-2 line. The FERMI FEL-2
beamline consists of 13 undulators of three different types. FEL-2 beamline
consists of a two-stage HGHG scheme, separated by a magnetic delay line. The
magnetic delay line is considered to make the fresh technique possible such
that by delay it can change the lasing window from the tail of the electron
beam to the head.

Fig. 6 summarizes the method and results of the superradiant experiment at
FERMI. Fig. 6(up) represents the three-stage layout of the experiment. In the
first modulator (M1) the seed interacted with the electron beam and induced an
energy modulation on it in the range of seed wavelength (264.4 nm). This
energy modulation was converted into the density modulation with components at
all the harmonics of the seed in the first dispersive section (DC1), and the
first two undulator sections (R1) were tuned into an additional frequency-
conversion stage at 88 nm (third harmonic of seed laser), to initiate the
cascade with a shorter gain length and longer slippage; then the next two
undulator sections (R2) were tuned to 44 nm radiation wavelength. The delay
line (DL) was used as a phase shifter to match the phase between the two
undulator sections. The dispersive section (DS2) was set to zero. The second-
stage radiator containing a chain of six undulators produced the FEL radiation
short pulse in 14 nm wavelength. We showed the evolution of pulse in the three
stages of the cascade both in Genesis [41] simulation and from the spectral
measurements done during the experiment with the spectrometer in PADRES
(Photon Analysis, Delivery, and Reduction System) [42, 43]. The first stage at
88 nm was added to start the cascade with higher gain (shorter gain length)
and longer slippage in the first sections. In this case, it was possible to
choose the right value of the seed peak power and of the first dispersive
section, to reach the saturation in the second undulator. Fig. 6(a) shows the
power profile (blue line) of the pulse and the slice energy spread (red line)
along the electron beam at the end of the second undulator as obtained in a
genesis simulation. We see that the pulse started to evolve in the
superradiant regime with a spike in the head growing above saturation. The
spectrum at the end of the second undulator in that stage consisted of two
symmetric sidebands that showed the distortion of the longitudinal profile. We
see a good agreement between simulation (Fig. 6(d)) and measurement (Fig.
6(g)) in regard to the shape of the spectrum. The saturation was associated
with an increase in the energy spread on the region of the beam, interacting
with the radiation. In the second stage, the bunching at the second harmonic
started to radiate at 44 nm. This emission was stronger on the head where the
bunching was higher and the energy spread was lower. On the contrary, the
radiation was strongly reduced in the tail by the energy spread accumulated in
the previous stage. This is shown in Fig. 6(b). Even at this stage, there was
a good agreement between the simulated spectrum and the measured one. Finally,
Fig. 6(c) shows the pulse profile and the energy spread at the end of the last
radiator. We see that the radiation was emitted by a small portion of the
head. The produced pulse was therefore a short spike with a peak power of 2 GW
and a pulse duration of 5 fs. The Spectrum profiles, both in simulation and in
measurement Fig. 6(f) and (l), showed a single pulse with some internal
modulation but sidebands did not appear.

The direct autocorrelation measurement of the FEL pulse duration was carried
out at the low-density matter (LDM) beamline. Autocorrelation in the EUV or
soft-X-ray regime is more difficult and less flexible than cross-correlation,
but for such short pulses, it is the most reasonable option. The measurement
was realized by splitting and recombining the FEL pulse with the FERMI split-
and-delay line autocorrelator/delay-creator and by monitoring the (energy-
resolved) signal of the two-photon above-threshold ionization of argon.

##  6 summary and conclusion

Well-controlled X-ray pulses with durations in the few-femtosecond and even
the sub-femtosecond range is essential in the investigation of dynamical
processes at the time scale of the motion of bound electrons in chemical,
physical and biological studies. Since soft-X-ray wavelengths can access core
electrons of extremely important light elements such as carbon, nitrogen, and
oxygen, fast soft-X-ray pulses can drive the system under study into a regime
where the transient excitation is not depleted by competing fast channels.
Novel free-electron lasers, producing ultrashort pulses with high peak power
from the extreme ultraviolet (EUV) to the soft-X-ray region have opened up a
wide range of new opportunities for scientists. In this review paper, we
described the progress and results of recent works on ultrafast pulse
generation and measurement of seed laser-assisted FEL pulse in the range of
EUV to soft X-Ray. We reported on the implementation of different approaches
in the FERMI facility. The pulse duration of the seeded FEL depends on the
seed laser pulse and longitudinal dispersion parameters, determining the
bunching factor at the beginning of the FEL radiator. Theoretically from the
bunching factor we can estimate the FEL pulse duration. The FERMI team
expressed the cross-correlation, autocorrelation, and SPIDER approaches at
seeded FEL and articulated the advantages and disadvantages of each method.
Basically, proper control of the seed laser pulse and electron beam parameters
governs the phase profile in the FEL pulse. By imposing a proper chirp on seed
laser frequency, we are able to shape the FEL pulse in both temporal and
spectral domains in either HGHG or EEHG performances. The implementation of
the chirp pulse amplification technique at seed FEL in the range of XUV showed
the three times compression and gave us hope that this technique can be
extended into the soft x-ray spectral window reducing the pulse duration even
below the femtosecond, with a very low transmission efficiency (0.2$\%$). The
pulse duration measurement in this experiment was executed by a cross-
correlation approach assisted by photo-emission of Helium gas. Furthermore, we
articulated the generation of a superradiant spike with the nonlinear
evolution approach. In this approach, we considered the cascade superradiant
process, and by employing the lower harmonic propagation, a small portion of
the electron beam was manipulated for nonlinear FEL amplification. The FERMI
team was able to generate a 5 fs FWHM pulse duration in 14.7 nm wavelength via
this technique. The pulse duration measurement in this experiment was carried
out by the autocorrelation method associated with the photo-emission of Argon
gas. For seeded FEL applications, all presented techniques promise new
possibilities for the FEL user community.

##  7 acknowledgments

The author expresses her sincere appreciation and gratitude to the FERMI team
for their exceptional contributions and pioneering work on short pulse
generation and superradiance experiments conducted at FERMI[40]. The author
would like to extend special acknowledgment for the fruitful discussions with
Luca Giannessi and Enrico Allaria.

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