# A short note on inf-sup conditions for the
Taylor-Hood family $Q_{k}$–$Q_{k-1}$

Walter Zulehner111Johann Radon Institute for Computational and Applied
Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz,
Austria. E-mail address<EMAIL_ADDRESS>
###### Abstract

We discuss two types of discrete inf-sup conditions for the Taylor-Hood family
$Q_{k}$–$Q_{k-1}$ for all $k\in\mathbb{N}$ with $k\geq 2$ in 2D and 3D. While
in 2D all results hold for a general class of hexahedral meshes, the results
in 3D are restricted to meshes of parallelepipeds. The analysis is based on an
element-wise technique as opposed to the widely used macroelement technique.
This leads to inf-sup conditions on each element of the subdivision as well as
to inf-sup conditions on the whole computational domain.

##  1 Introduction

Let $\mathcal{T}_{h}$ be a subdivision of a bounded polygonal/polyhedral
domain $\Omega\subset\mathbb{R}^{d}$, $d\in\\{2,3\\}$, into convex
quadrilaterals/hexahedrons $K$, which satisfy the standard compatibility
conditions. The mesh size of $\mathcal{T}_{h}$ is introduced by
$h=\max\\{h_{K}\colon K\in\mathcal{T}_{h}\\}$, where
$h_{K}=\operatorname{diam}K$ denotes the diameter of $K$. It is assumed that
each element $K\in\mathcal{T}_{h}$ is of the form

$K=F_{K}(\hat{K})$

with the reference element $\hat{K}=[0,1]^{d}$ and bijective and continuously
differentiable mappings $F_{K}\colon\hat{K}\to K$ for each
$K\in\mathcal{T}_{h}$. We assume the following standard shape regularity
conditions. There are positive constants $c_{i}$ such that

$c_{1}\,h_{K}^{d}\leq|J_{K}({\hat{x}})|\leq c_{2}\,h_{K}^{d}$

and

$c_{3}\,h_{K}^{2}\,I_{d}\leq J_{K}({\hat{x}})^{\top}J_{K}({\hat{x}})\leq
c_{4}\,h_{K}^{2}\,I_{d}$

for all ${\hat{x}}\in\hat{K}$ with the identity matrix $I_{d}$ in
$\mathbb{R}^{d}$. Here $J_{K}$ denotes the Jacobian of $F_{K}$ with
determinant $|J_{K}|$, for which we assume that $|J_{K}({\hat{x}})|>0$.

In particular, we will discuss meshes in 2D with $F_{K}\in Q_{1}(\hat{K})^{2}$
and in 3D with $F_{K}\in P_{1}(\hat{K})^{3}$ for all $K\in\mathcal{T}_{h}$.
Here we use the following standard notation of polynomial spaces: Let
$P_{k}(\hat{K})$ denote the set of polynomials on $\hat{K}$ with total degree
$\leq k$, and let $P_{k_{1},\ldots,k_{d}}(\hat{K})$ denote the set of
polynomials on $\hat{K}$ with degree $\leq k_{i}$ with respect to the variable
${\hat{x}}_{i}$, for $i=1,\ldots,d$. Finally, we set
$Q_{k}(\hat{K})=P_{k,\ldots,k}(\hat{K})$.

The family of generalized Taylor-Hood elements is given by the following pairs
of finite element spaces for $k\geq 2$:

$\displaystyle V_{h}$ $\displaystyle=\left\\{v\in H_{0}^{1}(\Omega)^{d}\colon
v\circ F_{K}\in Q_{k}(\hat{K})^{d}\ \text{for all}\
K\in\mathcal{T}_{h}\right\\}$ $\displaystyle Q_{h}$
$\displaystyle=\left\\{q\in L_{0}^{2}(\Omega)\cap C(\Omega)\colon q\circ
F_{K}\in Q_{k-1}(\hat{K})\ \text{for all}\ K\in\mathcal{T}_{h}\right\\},$

where $L_{0}^{2}(\Omega)=\\{q\in L^{2}(\Omega)\colon\int_{\Omega}q\ dx=0\\}$
with the standard Lebesgue space $L^{2}(\Omega)$, $H_{0}^{1}(\Omega)$ denotes
the set of functions in the standard Sobolev space $H^{1}(\Omega)$ which
vanish on $\partial\Omega$.

Let the bilinear form $b(v,q)$ be given by

$b(v,q)=-\int_{\Omega}q\,\operatorname{div}v\ dx.$

Then a first discrete inf-sup condition reads as follows:

$\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{H^{1}(\Omega)^{d}}}\gtrsim\|q_{h}\|_{L^{2}(\Omega)}>0\quad\text{for
all}\ q_{h}\in Q_{h}.$ (1)

Here and throughout the paper we use the following notation: $A\lesssim B$ if
there is a constant $c>0$ such that $A\leq c\,B$, $A\gtrsim B$ if $B\lesssim
A$, and later also $A\sim B$ if $A\lesssim B$ and $A\gtrsim B$. The involved
constants are meant to be independent of the mesh size $h$ (or the element
size $h_{K}$ for local estimates on some element $K$), but may depend on $k$
and the shape regularity constants $c_{i}$. This inf-sup condition is an
essential stability condition for, e.g., the discretization of the Stokes
problem.

We will also study a second type of discrete inf-sup condition:

$\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(\Omega)^{d}}}\gtrsim\|\nabla
q_{h}\|_{L^{2}(\Omega)^{d}}\quad\text{for all}\ q_{h}\in Q_{h}.$ (2)

This condition was discussed in [1], which turned out as an enabler of (2),
see [4]. Our interest for (2) is its role in the analysis of the smoothing
property of some multigrid method for the Stokes problem.

We have

$b(v,q)=\sum_{K\in\mathcal{T}_{h}}b_{K}(v,q)\quad\text{with}\quad
b_{K}(v,q)=-\int_{K}q\,\operatorname{div}v\ dx.$

Integration by parts results in the identity

$-\int_{K}q\,\operatorname{div}v\ dx=\int_{K}v\cdot\nabla q\ dx-\int_{\partial
K}v\cdot n\,q\ d\sigma$

for all $v\in H^{1}(K)$ and $q\in H^{1}(K)$. Here $n$ denotes the outward unit
normal vector to $K$. By the substitution rule we obtain

$\displaystyle-\int_{K}q\,\operatorname{div}v\ dx$
$\displaystyle=\int_{\hat{K}}\hat{v}_{K}\cdot
J_{K}^{-\top}\nabla\hat{q}_{K}\,|J_{K}|\
d{\hat{x}}-\int_{\partial\hat{K}}\hat{v}_{K}\cdot
J_{K}^{-\top}\hat{n}\,\hat{q}_{K}\,|J_{K}|d\hat{\sigma}$
$\displaystyle=\int_{\hat{K}}\hat{v}_{K}\cdot\operatorname{cof}(J_{K})\nabla\hat{q}_{K}\
d{\hat{x}}-\int_{\partial\hat{K}}\hat{v}_{K}\cdot\operatorname{cof}(J_{K})\hat{n}\,\hat{q}_{K}\
d{\hat{\sigma}}$ (3)

with $\hat{v}_{K}=v\circ F_{K}$, $\hat{q}_{K}=q\circ F_{K}$, and the outward
unit normal vector $\hat{n}$ to $\partial\hat{K}$. Here we used the
$|J|J^{-\top}=\operatorname{cof}(J)$, where $\operatorname{cof}(J)$ denotes
the cofactor matrix of a matrix $J$, see Appendix A.

The upcoming analysis relies on the following conditions for the integrands:

$\hat{v}_{K}\cdot\operatorname{cof}(J_{K})\nabla\hat{q}_{K}\in
Q_{2k-1}(\hat{K}),\quad\hat{v}_{K}\cdot\operatorname{cof}(J_{K})\hat{n}\,\hat{q}_{K}\in
Q_{2k-1}(\hat{f})$ (4)

on each edge/face $\hat{f}\subset\partial\hat{K}$ for $\hat{q}\in
Q_{k-1}(\hat{K})$ and $\hat{v}\in Q_{k}(\hat{K})^{d}$. These conditions allow
to represent the integrals in (3) by the tensor product $(k+1)$-point Gauss-
Lobatto quadrature rule exactly.

In 2D Condition (4) is well-known for general $F_{K}\in Q_{1}(\hat{K})^{2}$
and was essential for the proof of the inf-sup condition (1) on quadrilateral
meshes in [3]. In 3D it is easy to see that Condition (4) holds, if $F_{K}\in
P_{1}(\hat{K})^{3}$. Under this restriction the results in [3] easily extends
to 3D. A consequence of this restriction on $F_{K}$ is that in 3D only meshes
of parallelepipeds are covered by this note. A more detailed discussion of
Condition (4) can be found in Appendix B.

For simplicity the subindex $K$ in $F_{K}$, $J_{K}$, $\hat{v}_{K}$,
$\hat{q}_{K}$ is omitted if it is clear from the context, which $K$ is meant
like in sums over $K$.

##  2 $T$-coercivity

A standard technique to show an inf-sup condition is the macroelement
technique, see [2, 3]. A macroelement $M$ is a connected set of (a few)
elements $K\in\mathcal{T}_{h}$. An essential step of this technique is the
verification of an inf-sup condition for pairs $V_{M}$, $Q_{M}$ of local
spaces with

$V_{M}=\\{v|_{M}\colon v\in V_{h},\ v=0\ \text{on}\ \partial M\\},\quad
Q_{M}=\\{q|_{M}\colon q\in Q_{h}\\}$

associated to a collection of macroelements $M$ which cover $\Omega$.

In this paper we will consider instead the pairs

$V_{K}=\\{v|_{K}\colon v\in V_{h}\\},\quad Q_{K}=\\{q|_{K}\colon q\in
Q_{h}\\}$

associated to each element $K\in\mathcal{T}_{h}$. Note that functions in
$V_{K}$ do not necessarily vanish on $\partial K\setminus\partial\Omega$.

The following analysis was stimulated by the discussion in [5], where Taylor-
Hood elements for $k=2$ on meshes of mixed type consisting of hexahedrons,
prisms, pyramids, and tetrahedrons are considered, and where it is claimed
that, for all $K\in\mathcal{T}_{h}$, there is a linear mapping $T_{K}\colon
Q_{K}\to V_{K}$ such that

$b_{K}(T_{K}q_{h},q_{h})\gtrsim h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}\quad\text{and}\quad\|T_{K}q_{h}\|_{H^{1}(K)^{d}}\lesssim
h_{K}\,\|\nabla q_{h}\|_{L^{2}(K)^{d}}$ (5)

for all $q_{h}\in Q_{K}$, see [5, Lemma 3]. The proof was given only on the
reference element for $k=2$. In literature a property like (5) is often called
$T_{K}$-coercivity.

Here this result is extended to more general elements $K$ and to all $k\geq
2$.

######  Lemma 1.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $K\in\mathcal{T}_{h}$ and assume
that Condition (4) holds for $K$. Then Condition (5) holds.

###### Proof.

We will restrict ourselves to the 3D case, the 2D case is a simple adaptation
of the arguments used in 3D and is omitted.

As nodal degrees of freedom in $\hat{K}=[0,1]^{3}$ we choose the values at the
$(k+1)^{3}$ Gauss-Lobatto points, which are of the form
${\hat{\boldsymbol{a}}}=(x_{j_{1}},x_{j_{2}},x_{j_{3}})$, $1\leq j_{i}\leq
k+1$ and whose associated weights are of the form
$w_{\hat{\boldsymbol{a}}}=w_{j_{1}}\cdot w_{j_{2}}\cdot w_{j_{3}}$. Note that
$x_{1}=0$, $x_{k+1}=1$ and all weights are positive. The set of all nodes on
$\hat{K}$ are denoted by $\mathcal{N}(\hat{K})$.

Next we introduce the set of all nodal points in $K$ by
$\mathcal{N}(K)=\\{{\boldsymbol{a}}=F_{K}({\hat{\boldsymbol{a}}})\colon{\boldsymbol{a}}\in\mathcal{N}(\hat{K})\\}$,
the subset $\mathcal{N}(\mathring{K})$ denotes the set of all nodal points in
the interior of $K$. For $i\in\\{1,2,3\\}$, the set
$\mathcal{N}(\mathring{e}_{i})$ is formed by those nodal points
${\boldsymbol{a}}=F_{K}({\hat{\boldsymbol{a}}})$, where
${\hat{\boldsymbol{a}}}$ lies in the interior of edges of $\hat{K}$ parallel
to the ${\hat{x}}_{i}$-axis. For $i,j\in\\{1,2,3\\}$, $i\neq j$, the set
$\mathcal{N}(\mathring{f}_{\\{i,j\\}})$ is formed by those nodal points
${\boldsymbol{a}}=F_{K}({\hat{\boldsymbol{a}}})$, where
${\hat{\boldsymbol{a}}}$ lies in the interior of faces of $\hat{K}$ parallel
to the ${\hat{x}}_{i}$-axis as well as to the ${\hat{x}}_{j}$-axis. Finally,
to indicate that nodal points on the boundary $\partial\Omega$ are excluded we
write $\mathcal{N}^{*}(\cdot)$ instead of $\mathcal{N}(\cdot)$.

Let $q_{h}\in Q_{K}$. The nodal values of the element $v_{h}=T_{K}q_{h}\in
V_{K}$ are fixed on $K\in\mathcal{T}_{h}$ as follows.

$v^{({\boldsymbol{a}})}=\begin{cases}\left(J\nabla\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}(\mathring{K}),\\\\[8.61108pt]
\left(J_{i}\,\partial_{i}\hat{q}_{h}+J_{j}\,\partial_{j}\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}}),\\\\[8.61108pt]
\left(J_{i}\,\partial_{i}\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e_{i}}),\\\\[8.61108pt]
0&\quad\text{for all other}\ {\boldsymbol{a}}\in\mathcal{N}(K),\end{cases}$

with ${\hat{\boldsymbol{a}}}=F_{K}^{-1}({\boldsymbol{a}})$,
$\hat{q}_{h}=q_{h}\circ F_{K}$, the Jacobian $J$ of $F_{K}$, whose
$i^{\text{th}}$ column is denoted by $J_{i}$.

By using the property $J_{i}\cdot C_{j}=|J|\,\delta_{ij}$ with the Kronecker
symbol $\delta_{ij}$, where $C_{j}$ is the $j^{\text{th}}$ column of cofactor
matrix $C=\operatorname{cof}(J)$, it is easy to see that

$\left(\hat{v}_{h}\cdot\operatorname{cof}(J)\nabla\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})=\begin{cases}\left(|J|\|\nabla\hat{q}_{h}\|^{2}\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}(\mathring{K}),\\\\[8.61108pt]
\left(|J|(|\partial_{i}\hat{q}_{h}|^{2}+|\partial_{j}\hat{q}_{h}|^{2})\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}}),\\\\[8.61108pt]
\left(|J||\partial_{i}\hat{q}_{h}|^{2}\right)({\hat{\boldsymbol{a}}})&\quad\text{for}\
{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e_{i}}),\\\\[8.61108pt]
0&\quad\text{for all other}\ {\boldsymbol{a}}\in\mathcal{N}(K),\end{cases}$

and

$\left(\hat{v}_{h}\cdot\operatorname{cof}(J)\hat{n}\right)({\hat{\boldsymbol{a}}})=0\quad\text{for
all}\ {\hat{\boldsymbol{a}}}\in\partial\hat{K},$

with $\hat{v}_{h}=v_{h}\circ F_{K}$. Here $\|\cdot\|$ denotes the Euclidean
vector norm in $\mathbb{R}^{3}$.

By using Condition (4) and the exactness of the Gauss-Lobatto quadrature for
polynomials of degree $\leq 2k-1$ it follows that

$\displaystyle b_{K}(v_{h},q_{h})=-\int_{K}q_{h}\,\operatorname{div}v_{h}\ dx$
$\displaystyle\quad=\int_{\hat{K}}\hat{v}_{h}\cdot\operatorname{cof}(J)\nabla\hat{q}_{h}\
d{\hat{x}}-\int_{\partial\hat{K}}\hat{v}_{h}\cdot\operatorname{cof}(J)\hat{n}\,\hat{q}_{h}\
d{\hat{\sigma}}=\int_{\hat{K}}\hat{v}_{h}\cdot\operatorname{cof}(J)\nabla\hat{q}_{h}\
d{\hat{x}}$
$\displaystyle\quad=\sum_{{\boldsymbol{a}}\in\mathcal{N}(\mathring{K})}w_{\hat{\boldsymbol{a}}}\left(|J|\|\nabla\hat{q}_{h}\|^{2}\right)({\hat{\boldsymbol{a}}})+\sum_{\\{i,j\\}}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}})}w_{\hat{\boldsymbol{a}}}\left(|J|(|\partial_{i}\hat{q}_{h}|^{2}+|\partial_{j}\hat{q}_{h}|^{2})\right)({\hat{\boldsymbol{a}}})$
$\displaystyle\qquad\quad{}+\sum_{i}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e}_{i})}w_{\hat{\boldsymbol{a}}}\left(|J||\partial_{i}\hat{q}_{h}|^{2}\right)({\hat{\boldsymbol{a}}})$
$\displaystyle\quad\sim
h_{K}^{3}\,\bigg{(}\sum_{{\boldsymbol{a}}\in\mathcal{N}(\mathring{K})}\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}})+\sum_{\\{i,j\\}}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}})}\left(|\partial_{i}\hat{q}_{h}|^{2}+|\partial_{j}\hat{q}_{h}|^{2}\right)({\hat{\boldsymbol{a}}})$
$\displaystyle\qquad\quad\quad{}+\sum_{i}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e}_{i})}|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}})\bigg{)}.$
(6)

For the $L^{2}$-norm of $\nabla q_{h}$ on $K$ we obtain

$\displaystyle\|\nabla q_{h}\|_{L^{2}(K)^{3}}^{2}$
$\displaystyle=\int_{K}\|\nabla q_{h}\|^{2}\ dx\sim
h_{K}\,\int_{\hat{K}}\|\nabla\hat{q}_{h}\|^{2}\
d{\hat{x}}=h_{K}\,\sum_{{\boldsymbol{a}}\in\mathcal{N}(K)}w_{\hat{\boldsymbol{a}}}\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}})$
$\displaystyle\sim
h_{K}\,\sum_{{\boldsymbol{a}}\in\mathcal{N}(K)}\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}}).$
(7)

The expressions in (6) and (7) give rise to the introduction of two semi-norm
on $Q_{k-1}(\hat{K})$, given by

$\displaystyle|\hat{q}|_{1}^{2}$
$\displaystyle=\sum_{{\boldsymbol{a}}\in\mathcal{N}(\mathring{K})}\|\nabla\hat{q}\|^{2}({\hat{\boldsymbol{a}}})+\sum_{\\{i,j\\}}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}})}\left(|\partial_{i}\hat{q}|^{2}+|\partial_{j}\hat{q}|^{2}\right)({\hat{\boldsymbol{a}}})+\sum_{i}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e}_{i})}|\partial_{i}\hat{q}|^{2}({\hat{\boldsymbol{a}}}),$
$\displaystyle|\hat{q}|_{2}^{2}$
$\displaystyle=\sum_{{\boldsymbol{a}}\in\mathcal{N}(K)}\|\nabla\hat{q}\|^{2}({\hat{\boldsymbol{a}}}).$

Next we will study the kernels of these semi-norms. From the assumptions on
$K$ it follows that at least three adjacent faces of $K$ do not lie on the
boundary $\partial\Omega$. Without loss of generality we assume that these
three faces correspond to the faces ${\hat{x}}_{1}=0$, ${\hat{x}}_{2}=0$, and
${\hat{x}}_{3}=0$ on $\hat{K}$. Then all nodes from the sets

$\displaystyle\mathcal{N}_{1}$
$\displaystyle=\\{{\hat{\boldsymbol{a}}}=(x_{j_{1}},x_{j_{2}},x_{j_{3}})\in\mathcal{N}(\hat{K})\colon
2\leq j_{1}\leq k,\ 1\leq j_{2}\leq k,\ 1\leq j_{3}\leq k\\},$
$\displaystyle\mathcal{N}_{2}$
$\displaystyle=\\{{\hat{\boldsymbol{a}}}=(x_{j_{1}},x_{j_{2}},x_{j_{3}})\in\mathcal{N}(\hat{K})\colon
1\leq j_{1}\leq k,\ 2\leq j_{2}\leq k,\ 1\leq j_{3}\leq k\\},$
$\displaystyle\mathcal{N}_{3}$
$\displaystyle=\\{{\hat{\boldsymbol{a}}}=(x_{j_{1}},x_{j_{2}},x_{j_{3}})\in\mathcal{N}(\hat{K})\colon
1\leq j_{1}\leq k,\ 1\leq j_{2}\leq k,\ 2\leq j_{3}\leq k\\}$

correspond to nodes in the interior of $K$, or in the interior of some face,
or in the interior of some edge of $K$, but none of them corresponds to nodes
on the boundary $\partial\Omega$.

Assume now that $|\hat{q}|_{1}=0$ for some $\hat{q}\in Q_{k-1}(\hat{K})$. Then

$\partial_{1}\hat{q}({\hat{\boldsymbol{a}}})=0\quad\text{for all}\
{\hat{\boldsymbol{a}}}\in\mathcal{N}_{1}\quad\text{and}\quad\partial_{1}\hat{q}({\hat{\boldsymbol{a}}})\in
P_{k-2,k-1,k-1}(\hat{K}).$

Since $\mathcal{N}_{1}$ is a contiguous block of $(k-1)\cdot k\cdot k$ nodes,
where $\partial_{1}\hat{q}$ vanishes, it immediately follows that
$\partial_{1}\hat{q}$ vanishes everywhere. The same arguments apply to
$\partial_{2}\hat{q}$ and $\partial_{3}\hat{q}$ with the sets
$\mathcal{N}_{2}$ and $\mathcal{N}_{3}$, respectively, which eventually show
that $\nabla\hat{q}=0$, i.e. $\hat{q}$ is constant. So the kernel of the first
semi-norm is $Q_{0}(\hat{K})$. It is obvious that the kernel of the second
semi-norm is $Q_{0}(\hat{K})$, too. Since both semi-norms are defined on the
finite-dimensional space $Q_{k-1}(\hat{K})$ and have the same kernel, they are
equivalent. Therefore, we immediately obtain from (6) and (7) that

$b_{K}(v_{h},q_{h})\sim
h_{K}^{3}\,\|\nabla\hat{q}_{h}\|_{L^{2}(K)^{3}}^{2}\sim h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{3}}^{2}.$

In order to estimate the norms of $v_{h}$ observe that

$|v^{({\boldsymbol{a}})}|^{2}\lesssim
h_{K}^{2}\,\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}}),$

which implies

$\|v_{h}\|_{H^{1}(K)^{3}}^{2}\sim
h_{K}\,\sum_{{\boldsymbol{a}}\in\mathcal{N}(K)}|v^{({\boldsymbol{a}})}|^{2}\lesssim
h_{K}^{3}\sum_{{\boldsymbol{a}}\in\mathcal{N}(K)}\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}})\sim
h_{K}^{2}\,\|\nabla q_{h}\|_{L^{2}(K)^{3}}^{2}.$

This completes the proof. ∎

From this local result on $K$ we easily obtain the following global version.

######  Theorem 1.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $\mathcal{T}_{h}$ be a subdivision
of $\Omega$ such that Condition (4) holds for all $K\in\mathcal{T}_{h}$. Then
there is a linear mapping $T_{h}\colon Q_{h}\to V_{h}$ such that

$b(T_{h}q_{h},q_{h})\gtrsim\|\nabla
q_{h}\|_{h}^{2}\quad\text{and}\quad\|T_{h}q_{h}\|_{H^{1}(\Omega)^{d}}\lesssim\|\nabla
q_{h}\|_{h}\quad\text{for all}\ q_{h}\in Q_{h}$

with the mesh-dependent norm $\|.\|_{h}$, given by

$\|q_{h}\|_{h}^{2}=\sum_{K\in\mathcal{T}_{h}}h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}.$

###### Proof.

Let $q_{h}\in Q_{h}$. It is easy to see that the definition of the nodal
values of $v_{h}$ in the proof of Lemma 1 on common edges and faces of
neighboring elements is consistent, since the geometry mappings and $q_{h}$
are continuous across common edges and faces. This results in a well-defined
and continuous finite element function $v_{h}=T_{h}q_{h}\in V_{h}$, for which
we have

$\displaystyle b(v_{h},q_{h})$
$\displaystyle=\sum_{K\in\mathcal{T}_{h}}b(T_{K}q_{h},q_{h})\sim\sum_{K\in\mathcal{T}_{h}}h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}=\|\nabla q_{h}\|_{h}^{2},$ (8)

and

$\|v_{h}\|_{H^{1}(\Omega)^{d}}^{2}=\sum_{K\in\mathcal{T}_{h}}\|v_{h}\|_{H^{1}(K)^{d}}^{2}\lesssim\sum_{K\in\mathcal{T}_{h}}h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}=\|\nabla q_{h}\|_{h}^{2}.$

∎

The conditions on the bilinear form $b$ in Theorem 1 are called uniform
$T_{h}$-coercivity.

##  3 Inf-sup conditions

All results in this section hold for all $d\in\\{2,3\\}$ and all
$k\in\mathbb{N}$ with $k\geq 2$.

Immediate consequences of Lemma 1 and Theorem 1 are the following two results.

######  Corollary 1.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $K\in\mathcal{T}_{h}$ and assume
that Condition (4) holds for $K$. Then

$\sup_{0\neq v_{h}\in
V_{K}}\frac{b_{K}(v_{h},q_{h})}{\|v_{h}\|_{H^{1}(K)^{d}}}\gtrsim
h_{K}\,\|\nabla q_{h}\|_{L^{2}(K)^{d}}\quad\text{for all}\ q_{h}\in Q_{K}.$

######  Corollary 2.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $\mathcal{T}_{h}$ be a subdivision
of $\Omega$ such that Condition (4) holds for all $K\in\mathcal{T}_{h}$. Then

$\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{H^{1}(\Omega)^{d}}}\gtrsim\|q_{h}\|_{h}\quad\text{for
all}\ q_{h}\in Q_{h}.$

The classical inf-sup condition follows from Corollary 2 by applying the so-
called Verfürth trick, see [4, 2, 3, 5], to which we refer for a proof.

######  Theorem 2.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $\mathcal{T}_{h}$ be a subdivision
of $\Omega$ such that Condition (4) holds for all $K\in\mathcal{T}_{h}$. Then

$\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{H^{1}(\Omega)^{d}}}\gtrsim\|q_{h}\|_{L^{2}(\Omega)}\quad\text{for
all}\ q_{h}\in Q_{h}.$

Next we will discuss inf-sup conditions for a different pair of norms. First
we show a local version of [1, Proposition 1].

######  Corollary 3.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $K\in\mathcal{T}_{h}$ and assume
that Condition (4) holds for $K$. Then

$\sup_{0\neq v_{h}\in
V_{K}}\frac{b_{K}(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(K)^{d}}}\gtrsim\|\nabla
q_{h}\|_{L^{2}(K)^{d}}\quad\text{for all}\ q_{h}\in Q_{K}.$

###### Proof.

As in the proof of Lemma 1 one easily shows that

$\|T_{K}q_{h}\|_{L^{2}(K)^{d}}^{2}\lesssim h_{K}^{4}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}.$

Then

$\sup_{0\neq v_{h}\in
V_{K}}\frac{b_{K}(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(K)^{d}}}\geq\frac{b_{K}(T_{K}q_{h},q_{h})}{\|T_{K}q_{h}\|_{L^{2}(K)^{d}}}\gtrsim\frac{h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}^{2}}{h_{K}^{2}\,\|\nabla
q_{h}\|_{L^{2}(K)^{d}}}=\|\nabla q_{h}\|_{L^{2}(K)^{d}}.$

∎

The next result is a global version of Corollary 3 and can be seen as an
extension of [1, Proposition 1], where the proof was given for $k=2$ and for
meshes made of rectangles in 2D and bricks in 3D.

######  Theorem 3.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $\mathcal{T}_{h}$ be a subdivision
of $\Omega$ such that Condition (4) holds for all $K\in\mathcal{T}_{h}$. Then

$\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(\Omega)^{d}}}\gtrsim\|\nabla
q_{h}\|_{L^{2}(\Omega)^{d}}\quad\text{for all}\ q_{h}\in Q_{h}.$

###### Proof.

We will restrict ourselves to the 3D case, the 2D case is a simple adaptation
of the arguments used in 3D and is omitted.

Let $\mathcal{N}_{h}$ be the set of all nodes and let
$\varphi_{{\boldsymbol{a}}}$ be the nodal basis function associated to the
node ${\boldsymbol{a}}\in\mathcal{N}_{h}$. Then

$\|v_{h}\|_{L^{2}(\Omega)^{3}}^{2}\sim\sum_{{\boldsymbol{a}}\in\mathcal{N}_{h}}\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}^{2}\quad\text{for}\
v_{h}=\sum_{{\boldsymbol{a}}\in\mathcal{N}_{h}}v^{({\boldsymbol{a}})}\,\varphi_{{\boldsymbol{a}}}.$

For $q_{h}\in Q_{h}$ it follows that

$\displaystyle\left[\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(\Omega)^{3}}}\right]^{2}$
$\displaystyle\sim\left[\sup_{0\neq v_{h}\in
V_{h}}\frac{\sum_{{\boldsymbol{a}}\in\mathcal{N}_{h}}b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})}{\left(\sum_{{\boldsymbol{a}}\in\mathcal{N}_{h}}\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}^{2}\right)^{1/2}}\right]^{2}$
$\displaystyle=\sum_{{\boldsymbol{a}}\in\mathcal{N}_{h}}\left[\sup_{0\neq
v^{({\boldsymbol{a}})}\in\mathbb{R}^{3}}\frac{b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})}{\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}}\right]^{2}.$

We discuss three cases for the node ${\boldsymbol{a}}$.

  1. 1.

Assume that ${\boldsymbol{a}}\in\mathcal{N}(\mathring{K})$ for some
$K\in\mathcal{T}_{h}$. By setting

$v^{({\boldsymbol{a}})}=\left(J\nabla\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})$

we obtain as in the proof of Lemma 1

$b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})=b_{K}(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})\sim\,h_{K}^{3}\,\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}})$

and

$\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}^{2}=\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(K)^{3}}^{2}\sim
h_{K}^{3}\,|v^{({\boldsymbol{a}})}|^{2}\lesssim
h_{K}^{5}\,\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}}),$

which implies

$\left[\sup_{0\neq
v^{({\boldsymbol{a}})}\in\mathbb{R}^{3}}\frac{b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})}{\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}}\right]^{2}\gtrsim
h_{K}\,\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}}).$

  2. 2.

Next assume that ${\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e}_{i})$ for
some $K\in\mathcal{T}_{h}$. By setting

$v^{({\boldsymbol{a}})}=\left(J_{i}\,\partial_{i}\hat{q}_{h}\right)({\hat{\boldsymbol{a}}})$

we obtain as in the proof of Lemma 1

$b_{K}(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})\sim\,h_{K}^{3}\,|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}})$

and

$\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(K)^{3}}^{2}\lesssim
h_{K}^{5}\,|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}}).$

Analogous estimates follow for all other element $K^{\prime}$ which
${\boldsymbol{a}}$ belongs to. This leads to

$b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})=\sum_{K\in\mathcal{T}({\boldsymbol{a}})}b_{K}(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})\sim\sum_{K\in\mathcal{T}({\boldsymbol{a}})}h_{K}^{3}\,|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}})$

and

$\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}^{2}=\sum_{K\in\mathcal{T}({\boldsymbol{a}})}\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(K)^{3}}^{2}\lesssim\sum_{K\in\mathcal{T}({\boldsymbol{a}})}h_{K}^{5}\,|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}}),$

with
$\mathcal{T}({\boldsymbol{a}})=\\{K\in\mathcal{T}_{h}\colon{\boldsymbol{a}}\in
K\\}$. It easily follows that

$\left[\sup_{0\neq
v^{({\boldsymbol{a}})}\in\mathbb{R}^{3}}\frac{b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})}{\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}}\right]^{2}\gtrsim\sum_{K\in\mathcal{T}({\boldsymbol{a}})}h_{K}\,|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}}).$

  3. 3.

Finally, assume that
${\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}})$ for some
$K\in\mathcal{T}_{h}$. Then it follows completely analogously to the previous
case that

$\left[\sup_{0\neq
v^{({\boldsymbol{a}})}\in\mathbb{R}^{3}}\frac{b(v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}},q_{h})}{\|v^{({\boldsymbol{a}})}\varphi_{{\boldsymbol{a}}}\|_{L^{2}(\Omega)^{3}}}\right]^{2}\gtrsim\sum_{K\in\mathcal{T}({\boldsymbol{a}})}h_{K}\,\left(|\partial_{i}\hat{q}_{h}|^{2}+|\partial_{i}\hat{q}_{h}|^{2}\right)({\hat{\boldsymbol{a}}}).$

In summary, we have

$\displaystyle\left[\sup_{0\neq v_{h}\in
V_{h}}\frac{b(v_{h},q_{h})}{\|v_{h}\|_{L^{2}(\Omega)^{3}}}\right]^{2}$
$\displaystyle\quad\gtrsim\sum_{K\in\mathcal{T}_{h}}h_{K}\,\bigg{(}\sum_{{\boldsymbol{a}}\in\mathcal{N}(\mathring{K})}\|\nabla\hat{q}_{h}\|^{2}({\hat{\boldsymbol{a}}})+\sum_{\\{i,j\\}}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{f}_{\\{i,j\\}})}\left(|\partial_{i}\hat{q}_{h}|^{2}+|\partial_{j}\hat{q}_{h}|^{2}\right)({\hat{\boldsymbol{a}}})$
$\displaystyle\quad\qquad\quad\quad{}+\sum_{i}\sum_{{\boldsymbol{a}}\in\mathcal{N}^{*}(\mathring{e}_{i})}|\partial_{i}\hat{q}_{h}|^{2}({\hat{\boldsymbol{a}}})\bigg{)}\sim\sum_{K\in\mathcal{T}_{h}}\|\nabla
q_{h}\|_{L^{2}(K)^{3}}^{2}=\|\nabla q_{h}\|_{L^{2}(\Omega)^{3}}^{2},$

which completes the proof. ∎

##  Appendix A Some details on the cofactor matrix

For the convenience of the reader we collect here some well-known facts on the
cofactor matrix $C=\operatorname{cof}(J)$ of a matrix $J$. The elements of $C$
are given by $C_{ij}=(-1)^{i+j}M_{ij}$ with the minor $M_{ij}$, which is the
determinant of the submatrix of $J$ formed by deleting the $i^{\text{th}}$ row
and $j^{\text{th}}$ column. For a non-singular matrix $J$ it is known that

$(\det J)J^{-\top}=\operatorname{cof}(J).$

#### Cofactor matrix in 2D:

$\text{If}\quad J=\begin{pmatrix}J_{11}&J_{12}\\\
J_{21}&J_{22}\end{pmatrix},\quad\text{then}\quad\operatorname{cof}(J)=\begin{pmatrix}J_{22}&-J_{21}\\\
-J_{12}&J_{11}\end{pmatrix}.$

Note that the columns of the cofactor matrix differ from the columns of $J$ by
rotations by $\mp\pi/2$:

$\begin{pmatrix}J_{22}\\\ -J_{12}\end{pmatrix}=-\begin{pmatrix}J_{12}\\\
J_{22}\end{pmatrix}^{\bot},\quad\begin{pmatrix}-J_{21}\\\
J_{11}\end{pmatrix}=\begin{pmatrix}J_{11}\\\
J_{21}\end{pmatrix}^{\bot}\quad\text{with}\quad\begin{pmatrix}a\\\
b\end{pmatrix}^{\bot}=\begin{pmatrix}-b\\\ a\end{pmatrix}.$

If $J$ is the Jacobian of $F$, then

$\operatorname{cof}(J)=\left(-\left(\partial_{2}F\right)^{\bot}\ ,\
\left(\partial_{1}F\right)^{\bot}\right).$

#### Cofactor matrix in 3D:

If

$J=\begin{pmatrix}J_{11}&J_{12}&J_{13}\\\ J_{21}&J_{22}&J_{23}\\\
J_{31}&J_{32}&J_{33}\end{pmatrix},$

then

$\displaystyle\operatorname{cof}(J)$
$\displaystyle=\begin{pmatrix}J_{22}J_{33}-J_{32}J_{23}&-(J_{21}J_{33}-J_{31}J_{23})&J_{21}J_{32}-J_{31}J_{22}\\\
-(J_{12}J_{33}-J_{32}J_{13})&J_{11}J_{33}-J_{31}J_{13}&-(J_{11}J_{32}-J_{31}J_{12})\\\
J_{12}J_{23}-J_{22}J_{13}&-(J_{11}J_{23}-J_{21}J_{13})&J_{11}J_{22}-J_{21}J_{12}\end{pmatrix}$
$\displaystyle=\left(\begin{pmatrix}J_{12}\\\ J_{22}\\\
J_{32}\end{pmatrix}\times\begin{pmatrix}J_{13}\\\ J_{23}\\\
J_{33}\end{pmatrix}\,,\,\begin{pmatrix}J_{13}\\\ J_{23}\\\
J_{33}\end{pmatrix}\times\begin{pmatrix}J_{11}\\\ J_{21}\\\
J_{31}\end{pmatrix}\,,\,\begin{pmatrix}J_{11}\\\ J_{21}\\\
J_{31}\end{pmatrix}\times\begin{pmatrix}J_{12}\\\ J_{22}\\\
J_{32}\end{pmatrix}\right).$

If $J$ is the Jacobian of $F$, then

$\operatorname{cof}(J)=\left(\partial_{2}F\times\partial_{3}F\,,\,\partial_{3}F\times\partial_{1}F\,,\,\partial_{1}F\times\partial_{2}F\right).$

##  Appendix B Discussion of Condition (4)

We start the discussion by firstly collecting elementary properties of
$\operatorname{cof}(J)\nabla\hat{q}$.

######  Lemma 2.

Let $F\in Q_{1}(\hat{K})^{d}$ and $\hat{q}\in Q_{k-1}(\hat{K})$. Then we have

$\operatorname{cof}(J)\nabla\hat{q}\in\begin{cases}Q_{k-1}(\hat{K})^{2}&\quad\text{in
2D},\\\ Q_{k}(\hat{K})^{3}&\quad\text{in 3D},\end{cases}$

and

$\operatorname{cof}(J)\hat{n}\in\begin{cases}Q_{0}(\hat{f})^{2}&\quad\text{in
2D},\\\ Q_{1}(\hat{f})^{3}&\quad\text{in 3D},\end{cases}$

on each face $\hat{f}$ of $\hat{K}$.

###### Proof.

For later reference we recall a short proof.

In 2D the columns of the cofactor matrix have the following properties

$-\left(\partial_{2}F\right)^{\bot}\in
P_{1,0}(\hat{K})^{2},\quad\left(\partial_{1}F\right)^{\bot}\in
P_{0,1}(\hat{K})^{2}.$

Since

$\partial_{1}\hat{q}\in P_{k-2,k-1}(\hat{K}),\quad\partial_{2}\hat{q}\in
P_{k-1,k-2}(\hat{K}),$

it follows that

$\displaystyle\operatorname{cof}(J)\nabla\hat{q}$
$\displaystyle=-\partial_{1}\hat{q}\left(\partial_{2}F\right)^{\bot}+\partial_{2}\hat{q}\left(\partial_{1}F\right)^{\bot}$
$\displaystyle\in
P_{k-2,k-1}(\hat{K})P_{1,0}(\hat{K})^{2}+P_{k-1,k-2}(\hat{K})P_{0,1}(\hat{K})^{2}$
$\displaystyle\subset P_{k-1,k-1}(\hat{K})^{2}=Q_{k-1}(\hat{K})^{2}.$

For the edge $\hat{f}$ with ${\hat{x}}_{1}=0$, we have $\hat{n}=(-1,0)^{\top}$
and

$\operatorname{cof}(J)\hat{n}=-\left(\partial_{2}F\right)^{\bot}\in
Q_{0}(\hat{f})^{2}$

Therefore,

$\operatorname{cof}(J)\hat{n}\,\hat{q}\in Q_{k-1}(\hat{K}).$

Similar arguments hold for all other edges.

In 3D the columns of the cofactor matrix have the following properties

$\partial_{2}F\times\partial_{3}F\in
P_{2,1,1}(\hat{K})^{3},\quad\partial_{3}F\times\partial_{1}F\in
P_{1,2,1}(\hat{K})^{3},\quad\partial_{1}F\times\partial_{2}F\in
P_{1,1,2}(\hat{K})^{3}.$ (9)

Since

$\partial_{1}\hat{q}\in P_{k-2,k-1,k-1}(\hat{K}),\quad\partial_{2}\hat{q}\in
P_{k-1,k-2,k-1}(\hat{K}),\quad\partial_{3}\hat{q}\in
P_{k-1,k-1,k-2}(\hat{K}),$

it follows analogously to the 2D case that

$\displaystyle\operatorname{cof}(J)\nabla\hat{q}$
$\displaystyle=\partial_{1}\hat{q}\left(\partial_{2}F\times\partial_{3}F\right)+\partial_{2}\hat{q}\left(\partial_{3}F\times\partial_{1}F\right)+\partial_{3}\hat{q}\left(\partial_{1}F\times\partial_{2}F\right)$
$\displaystyle\in Q_{k}(\hat{K})^{3}.$

For the face $\hat{f}$ with ${\hat{x}}_{1}=0$, we have
$\hat{n}=(-1,0,0)^{\top}$ and

$\operatorname{cof}(J)\hat{n}=\partial_{2}F\times\partial_{3}F\in
P_{1,1}(\hat{f})^{3}=Q_{1}(\hat{f})^{3},$

Therefore

$\operatorname{cof}(J)\hat{n}\,\hat{q}\in Q_{k}(\hat{K}).$

Similar arguments hold for all other faces and for the 2D case.

∎

It is obvious that Lemma 2 implies Condition (4) in 2D. In order to ensure
Condition (4) also in 3D one would need

$\partial_{2}F\times\partial_{3}F\in
P_{1,0,0}(\hat{K})^{3},\quad\partial_{3}F\times\partial_{1}F\in
P_{0,1,0}(\hat{K})^{3},\quad\partial_{1}F\times\partial_{2}F\in
P_{0,0,1}(\hat{K})^{3}$ (10)

instead of (9).

######  Lemma 3.

Let $F\in Q_{1}(\hat{K})^{3}$. Then (10) holds if and only if $F\in
P_{1}(\hat{K})^{3}$.

###### Proof.

We have

$\displaystyle F({\hat{x}})$
$\displaystyle=(1-{\hat{x}}_{1})(1-{\hat{x}}_{2})(1-{\hat{x}}_{3})\,{\boldsymbol{a}}^{(1)}+{\hat{x}}_{1}(1-{\hat{x}}_{2})(1-{\hat{x}}_{3})\,{\boldsymbol{a}}^{(2)}$
$\displaystyle\quad{}+{\hat{x}}_{1}{\hat{x}}_{2}(1-{\hat{x}}_{3})\,{\boldsymbol{a}}^{(3)}+(1-{\hat{x}}_{1}){\hat{x}}_{2}(1-{\hat{x}}_{3})\,{\boldsymbol{a}}^{(4)}$
$\displaystyle\quad{}+(1-{\hat{x}}_{1})(1-{\hat{x}}_{2}){\hat{x}}_{3}\,{\boldsymbol{a}}^{(5)}+{\hat{x}}_{1}(1-{\hat{x}}_{2}){\hat{x}}_{3}\,{\boldsymbol{a}}^{(6)}$
$\displaystyle\quad{}+{\hat{x}}_{1}{\hat{x}}_{2}{\hat{x}}_{3}\,{\boldsymbol{a}}^{(7)}+(1-{\hat{x}}_{1}){\hat{x}}_{2}{\hat{x}}_{3}\,{\boldsymbol{a}}^{(8)}$

and

$\displaystyle\partial_{1}F({\hat{x}})$
$\displaystyle=(1-{\hat{x}}_{2})(1-{\hat{x}}_{3})\,({\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(1)})+{\hat{x}}_{2}(1-{\hat{x}}_{3})\,({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(4)})$
$\displaystyle\quad{}+(1-{\hat{x}}_{2}){\hat{x}}_{3}\,({\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(5)})+{\hat{x}}_{2}{\hat{x}}_{3}\,({\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(8)})$
$\displaystyle={\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(1)}+{\hat{x}}_{2}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(4)})+{\hat{x}}_{3}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(5)})$
$\displaystyle\quad{}+{\hat{x}}_{2}{\hat{x}}_{3}\,({\boldsymbol{a}}^{(2)}+{\boldsymbol{a}}^{(4)}+{\boldsymbol{a}}^{(5)}+{\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(1)}-{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(8)})$
$\displaystyle\partial_{2}F({\hat{x}})$
$\displaystyle=(1-{\hat{x}}_{1})(1-{\hat{x}}_{3})\,({\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(1)})+{\hat{x}}_{1}(1-{\hat{x}}_{3})\,({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)})$
$\displaystyle\quad{}+(1-{\hat{x}}_{1}){\hat{x}}_{3}\,({\boldsymbol{a}}^{(8)}-{\boldsymbol{a}}^{(5)})+{\hat{x}}_{1}{\hat{x}}_{3}\,({\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(6)})$
$\displaystyle={\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(1)}+{\hat{x}}_{1}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(4)})+{\hat{x}}_{3}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(8)}-{\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(5)})$
$\displaystyle\quad{}+{\hat{x}}_{1}{\hat{x}}_{3}\,({\boldsymbol{a}}^{(2)}+{\boldsymbol{a}}^{(4)}+{\boldsymbol{a}}^{(5)}+{\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(1)}-{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(8)})$
$\displaystyle\partial_{3}F({\hat{x}})$
$\displaystyle=(1-{\hat{x}}_{1})(1-{\hat{x}}_{2})\,({\boldsymbol{a}}^{(5)}-{\boldsymbol{a}}^{(1)})+{\hat{x}}_{1}(1-{\hat{x}}_{2})\,({\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(2)})$
$\displaystyle\quad{}+(1-{\hat{x}}_{1}){\hat{x}}_{2}\,({\boldsymbol{a}}^{(8)}-{\boldsymbol{a}}^{(4)})+{\hat{x}}_{1}{\hat{x}}_{2}\,({\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(3)})$
$\displaystyle={\boldsymbol{a}}^{(5)}-{\boldsymbol{a}}^{(1)}+{\hat{x}}_{1}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(5)})+{\hat{x}}_{2}\,({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(8)}-{\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(5)})$
$\displaystyle\quad{}+{\hat{x}}_{1}{\hat{x}}_{2}\,({\boldsymbol{a}}^{(2)}+{\boldsymbol{a}}^{(4)}+{\boldsymbol{a}}^{(5)}+{\boldsymbol{a}}^{(7)}-{\boldsymbol{a}}^{(1)}-{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(6)}-{\boldsymbol{a}}^{(8)})$

The coefficients $\operatorname{coeff}({\hat{x}}_{1})$ and
$\operatorname{coeff}({\hat{x}}_{2})$ of ${\hat{x}}_{1}$ and ${\hat{x}}_{2}$
of the polynomial

$\partial_{1}F({\hat{x}})\times\partial_{2}F({\hat{x}})$

are given by

$\displaystyle\operatorname{coeff}({\hat{x}}_{1})$
$\displaystyle=({\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(1)})\times({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(4)})=({\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(1)})\times({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(4)})$
$\displaystyle\operatorname{coeff}({\hat{x}}_{2})$
$\displaystyle=({\boldsymbol{a}}^{(1)}+{\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(4)})\times({\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(1)})=({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)})\times({\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(1)}).$

Therefore, the condition

$\partial_{1}F\times\partial_{2}F\in P_{0,0,1}(\hat{K})^{3}.$

leads to

$\displaystyle({\boldsymbol{a}}^{(2)}-{\boldsymbol{a}}^{(1)})\times({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(4)})$
$\displaystyle=0,$
$\displaystyle({\boldsymbol{a}}^{(3)}-{\boldsymbol{a}}^{(2)})\times({\boldsymbol{a}}^{(4)}-{\boldsymbol{a}}^{(1)})$
$\displaystyle=0.$

That means that opposite edges of the face spanned by the nodes
${\boldsymbol{a}}^{(1)}$, ${\boldsymbol{a}}^{(2)}$, ${\boldsymbol{a}}^{(3)}$,
${\boldsymbol{a}}^{(4)}$ are parallel, i.e., the face is a parallelogram.
Analogously, by discussing

$\partial_{2}F\times\partial_{3}F\quad\text{and}\quad\partial_{3}F\times\partial_{1}F$

it follows that the faces spanned by the nodes ${\boldsymbol{a}}^{(1)}$,
${\boldsymbol{a}}^{(4)}$, ${\boldsymbol{a}}^{(8)}$, ${\boldsymbol{a}}^{(5)}$
and by the nodes ${\boldsymbol{a}}^{(1)}$, ${\boldsymbol{a}}^{(2)}$,
${\boldsymbol{a}}^{(6)}$, ${\boldsymbol{a}}^{(5)}$ are parallelograms, too. By
the transformation ${\hat{x}}_{i}\mapsto 1-{\hat{x}}_{i}$ one obtains that the
remaining 3 faces of $K$ are parallelograms, too. Therefore, $K$ is a
parallelepiped and $F\in P_{1}(\hat{K})^{3}$. ∎

The following example of a hexahedron shows that Condition (4) does not hold
in 3D for general $F\in Q_{1}(\hat{K})^{3}$.

######  Example 1.

Consider the hexahedron with the nodes

${\boldsymbol{a}}^{(1)}=\begin{pmatrix}0\\\ 0\\\ 0\end{pmatrix},\
{\boldsymbol{a}}^{(2)}=\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix},\
{\boldsymbol{a}}^{(3)}=\begin{pmatrix}3/4\\\ 3/4\\\ 0\end{pmatrix},\
{\boldsymbol{a}}^{(4)}=\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}$
${\boldsymbol{a}}^{(5)}=\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix},\
{\boldsymbol{a}}^{(6)}=\begin{pmatrix}1/2\\\ 0\\\ 1\end{pmatrix},\
{\boldsymbol{a}}^{(7)}=\begin{pmatrix}3/8\\\ 3/8\\\ 1\end{pmatrix},\
{\boldsymbol{a}}^{(8)}=\begin{pmatrix}0\\\ 1/2\\\ 1\end{pmatrix}$

The geometry of the example is visualized in the following figure. In red: the
nodes of $K$. In blue: the (visible) nodes of $\hat{K}$.

For the particular choice of the nodes, one obtains by elementary calculations
that

$\partial_{2}F({\hat{x}})=\frac{{\hat{x}}_{3}-2}{8}\,\begin{pmatrix}{\hat{x}}_{1}\\\
{\hat{x}}_{1}-4\\\
0\end{pmatrix},\quad\partial_{3}F({\hat{x}})=\frac{1}{8}\,\begin{pmatrix}{\hat{x}}_{1}({\hat{x}}_{2}-4)\\\
{\hat{x}}_{2}({\hat{x}}_{1}-4)\\\ 8\end{pmatrix},$
$\partial_{2}F({\hat{x}})\times\partial_{3}F({\hat{x}})=\frac{{\hat{x}}_{3}-2}{16}\,\begin{pmatrix}2({\hat{x}}_{1}-4)\\\
-2{\hat{x}}_{1}\\\ {\hat{x}}_{1}({\hat{x}}_{1}-4)\end{pmatrix}.$

For

$\hat{v}({\hat{x}})=64\,{\hat{x}}_{1}(1-{\hat{x}}_{1}){\hat{x}}_{2}(1-{\hat{x}}_{2}){\hat{x}}_{3}(1-{\hat{x}}_{3})\,\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix}\in
Q_{2}(\hat{K})^{3},\quad\hat{q}({\hat{x}})={\hat{x}}_{1}\in Q_{1}(\hat{K})$

we obtain

$\displaystyle\hat{v}({\hat{x}})\cdot\operatorname{cof}(J({\hat{x}}))\nabla\hat{q}({\hat{x}})=\hat{v}({\hat{x}})\cdot\left(\partial_{2}F({\hat{x}})\times\partial_{3}F({\hat{x}})\right)$
$\displaystyle\quad=4\,{\hat{x}}_{1}^{2}\,(1-{\hat{x}}_{1})\,({\hat{x}}_{1}-4)\,{\hat{x}}_{2}(1-{\hat{x}}_{2}){\hat{x}}_{3}(1-{\hat{x}}_{3})({\hat{x}}_{3}-2)\notin
Q_{3}(\hat{K}),$

in contradiction to Condition (4) for $k=2$.

### Acknowledgements:

The author is very grateful for helpful information on the topic of the paper
provided by Andrea Bressan, Vivette Girault (communicated by Thomas Wick),
Rolf Rannacher (communicated by Thomas Wick), David Silvester, Rolf Stenberg,
and Christian Wieners.

## References

  * [1] M. Bercovier and O. Pironneau.  Error estimates for finite element method solution of the Stokes problem in the primitive variables.  Numer. Math., 33:211–224, 1979.
  * [2] Rolf Stenberg.  Analysis of mixed finite element methods for the Stokes problem: A unified approach.  Math. Comput., 42:9–23, 1984.
  * [3] Rolf Stenberg.  Error analysis of some finite element methods for the Stokes problem.  Math. Comput., 54(190):495–508, 1990.
  * [4] R. Verfürth.  Error estimates for a mixed finite element approximation of the Stokes equations.  RAIRO, Anal. Numér., 18:175–182, 1984.
  * [5] Christian Wieners.  Taylor-Hood elements in 3D.  In Analysis and simulation of multifield problems. Selected papers of the international conference on multifield problems, Stuttgart, Germany, April 8–10, 2002, pages 189–196. Berlin: Springer, 2003.


# Phase shift, Amplitude and Wavefunctions for np-system using Morse Potential
by Calogero’s Approach

Anil Khachi1,2

1Department of Physics and Astronomical Sciences,
Central University of Himachal Pradesh Dharamshala, 176215,
Himachal Pradesh, Bharat (India)

2 Department of Physics
St. Bedes College, 171002,
Himachal Pradesh, Bharat (India)

###### Abstract

Phase shift($\delta(r)$), Amplitude($A(r)$) and Wave function($u(r)$) vs r (in
fm) curves for various channels (S, P and D) of n-p scattering have been
calculated using phase function method (PFM) method. To do this, inverse
potentials obtained using the Morse function as the zeroth reference potential
is employed. Recently, the GRANADA group published a comprehensive partial
wave analysis of scattering data, consisting of 6713 np phase shift data
points from 1950 to 2013. Using the final experimental data points from
GRANADA we obtained the parameters for Morse potential by minimizing mean
square error (MSE) as the cost function. Various quantum functions i.e.
$\delta(r)$, $A(r)$ and $u(r)$ are described upto 5fm with energies $E_{\ell
ab}=[1,10,50,100,150,250,350]$.

keywords: n-p scattering, phase function method (PFM), Morse potential,
scattering phase shifts

##  1 Introduction

Under “standard approach” main objective of theoretical physicst is to
calculate the wavefunction from which other scattering properties like phase
shifts, low energy scattering parameters, static properties etc., can be
obtained. It is to be noted that the experimentalists do not get wavefunction
as an output of an experiment, rather one gets outputs like differential cross
sections and total cross-sections, transition energies etc., which are a
result of the interaction processes. Phase shifts are further related to the
cross sections. The interaction is portrayed from the experimental outputs in
form of V(r) vs r (fm) plots. According to Schwinger, Landau and Smorodinsky
[1] “The meeting place between theory and experiment is not the phase shifts
themselves but the value of the variational parameters implied by phase
shifts”.
There are various techniques that can be used to obtain the scattering phases
by solving the Schrödinger equation like: Born approximation, Brysk’s
approximation,and other successive approximation techniques. In our earlier
papers we have applied PFM or VPA for studying n-p [2, 3, 4], n-d [5] and
n-$\alpha$ [6] using Morse potential. Knowing that using PFM, phase shifts of
the nucleon-nucleon scattering can be obtained using different types of
potentials (high precision phenomenological potentials) and interaction models
between two nucleons. This paper is an extension of our recent published work
[8] where we have considered Morse potential [7] as nuclear interaction for
obtaining scattering phase shifts (SPS) for various $\ell$ channels in np
scattering. The current paper deals with using the obtained parameters [8] and
using those parameters we have obtained various functions like $\delta$ vs
r,$A$ vs r and $u$ vs r.

##  2 Methodology:

The Morse function is given by [7]

$V(r)=V_{0}(e^{-2\frac{(r-r_{m})}{a_{m}}}-2e^{-\frac{(r-r_{m})}{a_{m}}})$ (1)

In above equation, the parameters $V_{0}$, $r_{m}$, and $a_{m}$ represent the
strength of interaction between the particles, the equilibrium distance at
which maximum attraction is felt, and the shape of the potential,
respectively. The Morse potential possesses several distinguishing properties
that differentiate it from other phenomenological potentials, including: (i)
The Schr$\ddot{\text{o}}$dinger equation is exactly solvable for this
potential.(ii) Unlike other phenomenological potentials such as Manning-Rosen
[9], Malfliet-Tjon [10], Hulthén [11], and others, the Morse potential has an
exact analytical expression for the ${}^{1}S_{0}$ state.(iii) It exhibits a
relatively simpler wavefunction.(iv) It is a shape-invariant potential. Hence,
it can be considered as a good choice for modeling the interaction between any
two scattering particles.

###  2.1 Phase Function Method:

The Schr$\ddot{o}$dinger wave equation for a spinless particle with energy E
and orbital angular momentum $\ell$ undergoing scattering is given by

$\frac{\hbar^{2}}{2\mu}\bigg{[}\frac{d^{2}}{dr^{2}}+\big{(}k^{2}-\ell(\ell+1)/r^{2}\big{)}\bigg{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(2)

Where $k=\sqrt{E/(\hbar^{2}/2\mu)}$.

Second order differential equation Eq.2 has been transformed to the first
order non-homogeneous differential equation of Riccati type [12, 13] given by

$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{{k\left(\hbar^{2}/2\mu\right)}}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(3)

Prime denotes differentiation of phase shift with respect to distance and the
Riccati Hankel function of first kind is related to $\hat{j_{\ell}}(kr)$ and
$\hat{\eta_{\ell}}(kr)$ by
$\hat{h}_{\ell}(r)=-\hat{\eta}_{\ell}(r)+\textit{i}~{}\hat{j}_{\ell}(r)$ . In
integral form the above equation can be written as

$\delta(k,r)=\frac{-1}{{k\left(\hbar^{2}/2\mu\right)}}\int_{0}^{r}{V(r)}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j_{\ell}}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta_{\ell}}(kr)\bigg{]}^{2}dr$
(4)

Eq.3 is numerically solved using Runge-Kutta 5th order (RK-5) method [14] with
initial condition $\delta_{\ell}(0)=0$. For $\ell=0$, the Riccati-Bessel and
Riccati-Neumann functions $\hat{j}_{0}$ and $\hat{\eta}_{0}$ get simplified as
$\sin(kr)$ and $-\cos(kr)$, so Eq.3, for $\ell=0$ becomes

$\delta^{\prime}_{0}(k,r)=-\frac{V(r)}{{k\left(\hbar^{2}/2\mu\right)}}\sin^{2}[kr+\delta_{0}(k,r)]$
(5)

In above equations $k=\sqrt{E/(\hbar^{2}/2\mu)}$ and $\hbar^{2}/2\mu$ = 41.47
MeV fm2 for np case. In above equation the function $\delta_{0}(k,r)$ was
termed “Phase function” by Morse and Allis [15]. Similarly by varying the
Bessel functions for various l values by using following recurrence relations
[16]

$\hat{j}_{{\ell}+1}(kr)=\frac{2{\ell}+1}{kr}\hat{j}_{\ell}(kr)-\hat{j}_{{\ell}-1}(kr)\\\
$ (6)
$\hat{\eta}_{{\ell}+1}(kr)=\frac{2{\ell}+1}{kr}\hat{\eta}_{\ell}(kr)-\hat{\eta}_{{\ell}-1}(kr)$
(7)

we obtain PFM equation for P-wave having following form

$\delta_{1}^{\prime}(k,r)=\frac{-V(r)}{k\left(\hbar^{2}/2\mu\right)}\left[\frac{\sin\left(\delta_{1}+kr\right)-kr\cdot\cos\left(\delta_{1}+kr\right)}{kr}\right]^{2}$
(8)

PFM equation for D-wave takes following form

$\delta_{2}^{\prime}(k,r)=\frac{-V(r)}{k\left(\hbar^{2}/2\mu\right)})\left[-\sin\left(kr+\delta_{2}\right)-3\cos\frac{\left(\delta_{2}+kr\right)}{kr}+3\sin\frac{\left(\delta_{2}+kr\right)}{(kr)^{2}}\right]^{2}$
(9)

The equation for amplitude function[17] with initial condition is obtained in
the form

$\displaystyle A_{\ell}^{\prime}(r)=$
$\displaystyle-\frac{A_{\ell}V(r)}{k}\left[\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$
(10)
$\displaystyle\times\left[\sin(\delta_{\ell}(k,r))(\hat{j}_{\ell}(kr)+\cos(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$

also the equation to obtained wavefunction[17] is

$u_{\ell}(r)=A_{\ell}(r)\left[\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$
(11)

Figure 1: Detailed flowchart for obtaining phase shift, amplitude function and
wavefunction.

##  3 Results and Discussion

Table 1: Model Parameters for Morse potential for various channels of n-p
scattering [8].

States | $V_{0}(MeV)$ | $r_{m}(fm)$ | $a_{m}(fm)$ | $MSE$
---|---|---|---|---
${}^{1}S_{0}$ | 70.438 | 0.901 | 0.372 | 0.649
${}^{1}P_{1}$ | 0.010 | 5.442 | 1.016 | 1.568
${}^{3}P_{0}$ | 11.579 | 1.750 | 0.601 | 0.049
${}^{3}P_{1}$ | 0.010 | 4.514 | 0.778 | 0.832
${}^{1}D_{2}$ | 131.302 | 0.010 | 0.526 | 0.026
${}^{3}D_{2}$ | 106.379 | 0.209 | 0.747 | 0.066

Figure 2: The potential $V(r)$, scattering phase shift $\delta(r)$, amplitude
$A(r)$, and wavefunction $u(r)$ for the ${}^{1}S_{0}$ state are shown in
Figure 2. The dashed line represents the potential obtained from the JISP
model [18, 19], while the dotted lines correspond to the NIJM-II potential
from [19]. Figure 3: The potential $V(r)$, scattering phase shift $\delta(r)$,
amplitude $A(r)$, and wavefunction $u(r)$ for the ${}^{1}P_{1}$ state are
shown in Figure 3. The dashed lines correspond to the NIJM-II potential [19].
Figure 4: The potential $V(r)$, scattering phase shift $\delta(r)$, amplitude
$A(r)$, and wavefunction $u(r)$ for the ${}^{3}P_{0}$ state are shown in
Figure 4. The dashed line represents the potential obtained from the JISP
model [18, 19], while the dotted lines correspond to the NIJM-II potential
[19]. Figure 5: The potential $V(r)$, scattering phase shift $\delta(r)$,
amplitude $A(r)$, and wavefunction $u(r)$ for the ${}^{3}P_{1}$ state are
shown in Figure 5. The dashed line represents the potential obtained from the
JISP model [18, 19], while the dotted lines correspond to the NIJM-II
potential [19]. Figure 6: The potential $V(r)$, scattering phase shift
$\delta(r)$, amplitude $A(r)$, and wavefunction $u(r)$ for the ${}^{1}D_{2}$
state are shown in Figure 6. The dashed line represents the potential obtained
from the JISP model [18, 19], while the dotted lines correspond to the NIJM-II
potential [19]. Figure 7: The potential $V(r)$, scattering phase shift
$\delta(r)$, amplitude $A(r)$, and wavefunction $u(r)$ for the ${}^{3}D2$
state are shown in Figure 7. The dashed lines correspond to the NIJM-II
potential [19].

###  3.1 Discussion

  1. 1.

For ${}^{1}S_{0}$ state in the region $0\leq r\leq 1$ fm, low-energy particles
face the attractive part of the potential. As the energy increases to $350$
MeV $(0\leq r\leq 1$ fm), the repulsive interaction takes place, indicating
that the interaction is occurring near the repulsive core. According to the
PFM equation, the repulsive potential will result in negative phase shifts.
This phenomenon is clearly depicted in Figure 2(b). For $r\geq 1$ fm, the
interaction moves towards the attractive core of the potential, leading to
positive phase shifts. The amplitude variation for the ${}^{1}S_{0}$ state is
shown in Figure2(c) and exhibits a very good match with those observed in
Zhaba’s work [17] using Av-18 potential [5]. Finally, the wave functions are
plotted at various energies ($E=10$, $50$, $150$, and $250$ MeV). At $10$ MeV,
our obtained wave function ([14, 15] is not observed to be in good aggreement
and can be seen for energies greater than $50$ MeV.

  2. 2.

For ${}^{1}P_{1}$ state, the potential is purely repulsive in nature. From PFM
(Phase Function Method), we know that a positive potential will lead to
negative phase shifts. The variation of phase shifts can be observed to be
very small for lower energies because, at lower energies, the interaction is
away from the repulsive core and the effective interaction is very close to
zero. However, as the energy increases, the repulsive core interaction comes
into play, resulting in a significant negative phase shift. The variation of
the amplitude function also exhibits reasonable variation with the NIJM-II
[15].

  3. 3.

${}^{3}P_{0}$ potential is having both attractive and repulsive nature. Hence
phase shift variation from negative to positive is observed in Figure 4.
Almost identical variation for amplitudes at different energies is observed,
which indicates that our inverse potential is in good relation with that of
Av-18. A small magnitude difference between our and Zhaba’s amplitude is
found, which may be a result of a small difference between the depths of Morse
and Av-18 potential. Again, the wave function for the 3P0 state is found to be
in very good match with those by NIJM-II [15] and JISP [14]. Our wave function
can be seen to be in close agreement with those given by NIJM-II.

  4. 4.

As in the ${}^{1}P_{1}$ case, for the ${}^{3}P_{1}$ state, the potential is
purely repulsive in nature, as shown in Figure 5(a). From PFM, we know that a
positive potential will lead to negative phase shifts, as seen in Figure 5(b).
The variation of phase shifts can be observed to be very small for lower
energies because, at lower energies, the interaction is away from the
repulsive core and the effective interaction is very close to zero. However,
as the energy increases, the repulsive core interaction comes into play,
resulting in a significant negative phase shift. The variation of the
amplitude function also exhibits reasonable variation with NIJM-II [15] .

  5. 5.

For the ${}^{1}D_{2}$ state, the potential is having a Gaussian shape, i.e.,
purely attractive with a tail going to zero. This indicates that the phase
shift must be purely positive, as shown in Figure 6(b). The phase shift is
again in good match with that given by Zhaba. The wavefunction for the 1D2
state is in excellent agreement with that given by the Av-18 potential and
JISM method, as shown in Figure 6(d).

  6. 6.

For the ${}^{3}D_{2}$ state, the potential is also seen to be purely
attractive. The phase shift, amplitudes, and wavefunction variation are shown
in Figure 7(a), Figure 7(b), and Figure 7(c), respectively.

Overall, the results indicate that our inverse procedure is validated by
obtaining the wavefunctions in close agreement with the Av-18 potential and
JISP. However, a certain phase lag is observed between our results and those
of NIJM-II and JISP [18, 19] for certain states. This phase lag can be
attributed to the simplistic Morse model used in our approach. To refine the
results, we can apply the Combinatorial Data Analysis (CDA) procedure
developed by our group [4]. This procedure can generate a certain class of
isospectral potentials that may provide better agreement between the compared
wavefunctions in this work.

##  4 Conclusion

The obtained phases serve as a quantum mechanical magnification glass,
allowing us to explore the intricacies of the nuclear domain. The variation of
phase shift with distance provides a beautiful demonstration that, for
repulsive potentials, the phase shift $\delta$ takes on positive values, while
for attractive potentials, it assumes negative values. This observation aligns
perfectly with the main equation of the Phase Function Method (PFM). The
presence of Bessel functions in the PFM equation leads to the oscillatory
nature of the phase shift for certain states. Our inverse procedure is
effective in obtaining Phase shift($\delta(r)$), Amplitude($A(r)$) and Wave
function($u(r)$) for various quantum mechanical states.

The wave function for specific states, such as ${}^{3}P_{0}$, ${}^{3}P_{1}$,
and ${}^{1}D_{2}$, exhibits excellent agreement with those obtained by NIJM-II
and JISP [18, 19]. This computational approach to constructing inverse
potentials offers certain advantages, although further improvements are
desired for enhanced performance. Moreover, this methodology can be extended
to investigate a wide range of scattering scenarios, including $n-\alpha$,
$p-\alpha$, $n-C$ among others. We are actively exploring these possibilities.

##  5 Acknowledgements

I am grateful to Prof. O. S. K. S. Sastri, my PhD advisor, my colleague Lalit
Kumar, and M.Sc. student Sudhir Banyal for their insightful remarks and
suggestions on this study.

## Appendix

### Amplitude Function Equations

for $\ell=0$,$\ell=1$, $\ell=2$

$\displaystyle A_{0}^{\prime}$
$\displaystyle=-\frac{A_{0}V(r)}{k\left(\frac{\hbar^{2}}{2\mu}\right)}\left[\cos\delta_{0}\cdot\sin(kr)-\sin\delta_{0}\cdot(-\cos(kr))\right]$
$\displaystyle\quad\times\left[\sin\delta_{0}\cdot\sin(kr)+\cos\delta_{0}\cdot(-\cos(kr))\right]$
(12) $\displaystyle A_{1}^{\prime}$
$\displaystyle=-\frac{A_{1}V(r)}{k\left(\frac{\hbar^{2}}{2\mu}\right)}\left[\cos\delta_{1}\left(\frac{\sin(kr)}{(kr)}-\cos(kr)\right)-\sin\delta_{1}\left(-\frac{\cos(kr)}{(kr)}-\sin(kr)\right)\right]$
$\displaystyle\quad\times\left[\sin\delta_{1}\left(\frac{\sin(kr)}{(kr)}-\cos(kr)\right)+\cos\delta_{1}\left(-\frac{\cos(kr)}{(kr)}-\sin(kr)\right)\right]$
(14) $\displaystyle A_{2}^{\prime}$
$\displaystyle=-\frac{A_{2}V(r)}{k\left(\frac{\hbar^{2}}{2\mu}\right)}\biggl{[}\cos\delta_{2}\left(\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\right)$
$\displaystyle\quad\times\left(\sin\delta_{2}\left(\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\right)+\cos\delta_{2}\left(\left(-\frac{3}{(kr)^{2}}+1\right)\cos(kr)\right.\right.$
$\displaystyle\quad\left.\left.-\frac{3}{(kr)}\sin(kr)\right)\right]$ (15)

### Wavefunction Equations

for $\ell=0$, $\ell=1$, $\ell=2$

$\displaystyle
u_{0}(r)=A_{0}(r)\left[\cos\delta_{0}(r)\cdot\sin(kr)-\sin\delta_{0}(r)\cdot\cos(kr)\right]$
(16) $\displaystyle
u_{1}(r)=A_{1}(r)\left[\cos\delta_{1}(r)\left(\frac{\sin(kr)}{(kr)}-\cos(kr)\right)-\sin\delta_{1}(r)\left(-\frac{\cos(kr)}{(kr)}-\sin(kr)\right)\right]$
(17) $\displaystyle
u_{2}(r)=A_{2}(r)\bigg{[}\cos\delta_{2}(r)\bigg{(}\left(\frac{3}{(kr)^{2}}-1\right)\sin(kr)-\frac{3}{(kr)}\cos(kr)\bigg{)}$
$\displaystyle\qquad-\sin\delta_{2}(r)\bigg{(}-\left(-\frac{3}{(kr)^{2}}+1\right)\cos(kr)-\frac{3}{(kr)}\sin(kr)\bigg{)}\bigg{]}$
(18)

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