# Fully discrete pointwse smoothing error estimates for measure valued initial
data

Dmitriy Leykekhman  Department of Mathematics, University of Connecticut,
Storrs, CT 06269, USA (dmitriy.leykekhman@uconn.edu).  ,  Boris Vexler  Chair
of Optimal Control, Technical University of Munich, School of Computation
Information and Technology, Department of Mathematics , Boltzmannstraße 3,
85748 Garching b. Munich, Germany (vexler@tum.de).  and  Jakob Wagner  Chair
of Optimal Control, Technical University of Munich, School of Computation
Information and Technology, Department of Mathematics , Boltzmannstraße 3,
85748 Garching b. Munich, Germany<EMAIL_ADDRESS>ORCID:
https://orcid.org/0000-0001-8510-9790.

###### Abstract.

In this paper we analyze a homogeneous parabolic problem with initial data in
the space of regular Borel measures. The problem is discretized in time with a
discontinuous Galerkin scheme of arbitrary degree and in space with continuous
finite elements of orders one or two. We show parabolic smoothing results for
the continuous, semidiscrete and fully discrete problems. Our main results are
interior $L^{\infty}$ error estimates for the evaluation at the endtime, in
cases where the initial data is supported in a subdomain. In order to obtain
these, we additionally show interior $L^{\infty}$ error estimates for $L^{2}$
initial data and quadratic finite elements, which extends the corresponding
result previously established by the authors for linear finite elements.

###### Key words and phrases:

optimal control, sparse control, initial data identification, smoothing
estimates, parabolic problems, finite elements, discontinuous Galerkin, error
estimates, pointwise error estimates

###### 1991 Mathematics Subject Classification:

65N30,65N15

##  1\. Introduction

In this work we discuss smoothing properties of the fully discrete
approximation of the homogeneous parabolic problem

$\displaystyle\partial_{t}v-\Delta v$ $\displaystyle=0$
$\displaystyle\text{in}\quad(0,T)\times\Omega,\;$ (1) $\displaystyle v$
$\displaystyle=0$ $\displaystyle\text{on}\quad(0,T)\times\partial\Omega,$
$\displaystyle v(0)$ $\displaystyle=v_{0}$
$\displaystyle\text{in}\quad\Omega,$

where $\Omega\subset\mathbb{R}^{N}$, $N=2,3$ is a bounded, convex,
polygonal/polyhedral domain. In particular we are interested in pointwise
error estimates in the case when the initial condition $v_{0}$ is a regular
Borel measure $v_{0}\in\mathcal{M}(\Omega)$ supported in some subdomain
$\Omega_{0}$ such that $\overline{\Omega}_{0}\subset\Omega$, for example a
linear combination of Dirac delta functions,
$v_{0}=\sum_{j}\beta_{j}\delta_{x_{j}}$. Our main result of this paper
establishes the fully discrete error estimate of the form

$\|(v-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(\Omega_{0},T)\left({k^{2r+1}}+\ell_{kh}{h^{s+1}}\right)\|{v_{0}}\|_{\mathcal{M}(\Omega)}.$
(2)

Here $r\geq 0$ is the order of time discretization, $s=1,2$ is the order of
the space discretization, and $\ell_{kh}$ is a logarithmic term that depends
on the mesh size $h$ and the maximum time step $k$. In order to simplify the
presentation, we assume that $v_{0}$ is supported in the same subdomain
$\Omega_{0}$, in which the $L^{\infty}$ error is estimated, wheareas in
general, two different subdomains could be chosen. We would like to point out
that the piecewise linear case $s=1$ does not require any additional
smoothness assumptions beyond regularity results available on convex domains
$\Omega$. The higher order convergence of $s=2$ requires some additional
smoothness assumptions, which are available for example on rectangular domains
(cf. Section 6). In this case, the logarithmic term $\ell_{kh}$ only depends
on $k$.

The above problem is classical and many important results are available in the
literature. The $L^{2}$ theory for a uniform time partition is well presented
in the classical textbook of Thomeé [18]. Extensions to variable time steps
are available in Eriksson et al. [7]. The $L^{1}\to L^{\infty}$ stability
results, are technically more difficult and one of the first papers in this
direction was the work of Schatz et al. [16], where such results were
established in two space dimensions for piecewise linear elements and strongly
A-stable single step methods with uniform time steps. The sharpest result in
the case of smooth domains and uniform time steps was obtained by A. Hansbo in
[9].

In our previous paper [14], for piecewise linear space discretizations on a
convex polygonal/polyhedral domain $\Omega$ and $v_{0}\in L^{2}(\Omega)$, we
have obtained

$\|(v-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T)\left({k^{2r+1}}+\ell_{kh}{h^{2}}\right)\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)},$ (3)

with explicit form of the constant $C(T)$. Such results were required for
obtaining sharp results in initial data estimation of the parabolic problems
from final time observation [14]. However, in order to extend the results to
the situation when the final time observation is taken at a finite number of
points [13], which is more relevant in applications, we require the results of
the form (2). This yields an error estimate for the adjoint state, which
satisfies a backwards-in-time problem, with a final time condition given by a
measure supported in the observation points. Since these points are fixed,
this support is contained in a proper subdomain, and hence the assumptions for
(2) are satisfied. In summary, the main contribution of our paper is the
establishment of fully discrete error estimates (2) for Galerkin methods on
potentially highly varying time partitions and quasi-uniform meshes on convex
polygonal/polyhedral domains, without any additional smoothness assumptions in
the case of piecewise linear case and with additional smoothness assumptions
in the case of quadratic elements.

The rest of the paper is organized as follows. In Section 2, we review the
notion of very weak solutions for parabolic homogeneous problems with initial
data given in the space of regular Borel measures. In Section 3, we discuss
space-time discretization schemes and introduce the semidiscrete and fully
discrete Galerkin solutions to (1). In Section 4, we review and show
continuous and discrete smoothing estimates for the continuous, semidiscrete
and fully discrete solutions. Our main result will be the pointwise fully
discrete error estimate for initial data in $\mathcal{M}(\Omega)$, see Theorem
5.2, which we establish in Section 5. Finally, in Section 6 we extend our main
result to a higher order space discretization.

##  2\. Very weak solutions and regularity

We begin by introducing the proper setup for the existence and regularity of
the solution with measure valued initial data. Let
$v_{0}\in\mathcal{M}(\Omega)$ be given. A function $v\in
L^{1}((0,T)\times\Omega)$ is called a very weak solution to the heat equation
(1), if

$-\int_{(0,T)\times\Omega}\left(\frac{\partial\varphi}{\partial
t}+\Delta\varphi\right)v\ dx\ dt=\int_{\Omega}\varphi(0)\
dv_{0}\qquad\text{for all }\varphi\in\Phi_{T},$ (4)

where the space $\Phi_{T}$ of all test functions is given by

$\Phi_{T}=\left\\{\varphi\in\Phi:\ \varphi(T)=0\ \text{in
}\Omega\right\\}\quad\text{with}\quad\Phi=\left\\{\varphi\in
L^{2}(0,T;H^{1}_{0}(\Omega)):\ \frac{\partial\varphi}{\partial
t}+\Delta\varphi\in L^{\infty}((0,T)\times\Omega)\land\varphi(T)\in
L^{2}(\Omega)\right\\}.$ (5)

With this definition, we have the following result (see [5, Lemma 2.2]):

######  Theorem 2.1.

For a given $v_{0}\in\mathcal{M}(\Omega)$, there exists a unique solution $v$
in the sense of (4). The solution $v$ lies in the space
$L^{r}(0,T;W^{1,p}_{0}(\Omega))$ for any $p,r\in[1,2)$ with
$\frac{2}{r}+\frac{N}{p}>N+1$, with the estimate

$\|v\|_{L^{r}(0,T;W^{1,p}_{0}(\Omega))}\leq C_{r,p}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}.$

Moreover, $v\in C([0,T];W^{-1,p}(\Omega))$, making the evaluation $v(t)$ well
defined at any $t\in[0,T]$, and in addition $v(T)\in L^{2}(\Omega)$ with

$\left\lVert v(T)\right\rVert_{L^{2}(\Omega)}\leq C_{T}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}.$ (6)

For any $\varphi\in\Phi$ there holds

$\int_{\Omega}\varphi(T)v(T)-\int_{(0,T)\times\Omega}\left(\frac{\partial\varphi}{\partial
t}+\Delta\varphi\right)v\ dx\ dt=\int_{\Omega}\varphi(0)\ dv_{0}.$ (7)

In the second estimate of the above theorem, the constant $C_{T}$ depends on
$T$. We shall make this dependence more explicit in Lemma 4.2. For the error
analysis below, we will require the following result.

######  Lemma 2.2.

Let $\tau\in(0,T)$ and let $v_{1}$ be the very weak solution of the heat
equation on the subinterval $(0,\tau)$ in the sense of (4). Let $v_{2}$ be the
weak solution of the heat equation on the subinterval $(\tau,T)$ with initial
data $v_{1}(\tau)\in L^{2}(\Omega)$. Then $v$ defined as

$v(t)=\left\\{\begin{array}[]{ll}v_{1}(t)&t\in(0,\tau]\\\
v_{2}(t)&t\in(\tau,T)\end{array}\right.$ (8)

is the very weak solution in the sense of (4).

######  Proof 2.3.

The proof is straightforward.

##  3\. Discretization

In this section we describe the semidiscrete and fully discrete finite element
discretizations of the homogeneous equation (1) and present smoothing type
error estimates. To discretize the problem, we use continuous Lagrange finite
elements of order $s\geq 1$ in space and discontinuous Galerkin methods of
order $r\geq 0$ in time.

###  3.1. Time discretization

To be more precise, we partition $I=(0,T]$ into subintervals
$I_{m}=(t_{m-1},t_{m}]$ of length $k_{m}=t_{m}-t_{m-1}$, where
$0=t_{0}<t_{1}<\cdots<t_{M-1}<t_{M}=T$. The maximal and minimal time steps are
denoted by $k=\max_{m}k_{m}$ and $k_{\min}=\min_{m}k_{m}$, respectively. We
impose the following conditions on the time mesh (as in [11] or [15]):

  1. (i)

There are constants $c,\beta>0$ independent on $k$ such that

$k_{\min}\geq ck^{\beta}.$

  2. (ii)

There is a constant $\kappa>0$ independent on $k$ such that for all
$m=1,2,\dots,M-1$

$\kappa^{-1}\leq\frac{k_{m}}{k_{m+1}}\leq\kappa.$

  3. (iii)

It holds $k\leq\frac{T}{2r+2}$.

The semidiscrete space $X_{k}^{r}$ for the case $v_{0}\in L^{2}(\Omega)$ is
taken as

$X_{k}^{r}=\Set{\varphi_{k}\in
L^{2}(I;H^{1}_{0}(\Omega))}{\varphi_{k}|_{I_{m}}\in\mathbb{P}_{r}(I_{m};H^{1}_{0}(\Omega)),\
m=1,2,\dots,M},$

where $\mathbb{P}_{r}(I_{m};V)$ is the space of polynomial functions of degree
$r$ in time on $I_{m}$ with values in a Banach space $V$. However, for the
semidiscrete formulation of (1) with measure valued initial data
$v_{0}\in\mathcal{M}(\Omega)$ to be well defined, at initial time the test
functions $\varphi_{k}$ need to be in $C_{0}(\Omega)$. Since for $N\geq 2$ the
space $H^{1}_{0}(\Omega)$ is not embedded in the space of continuous
functions, we need to modify the spaces of trial and test functions, as

$\widetilde{X}_{k}^{r}=\left\\{\varphi_{k}\in
L^{2}(I\times\Omega):\varphi_{k}|_{I_{m}}\in\mathbb{P}_{r}(I_{m};W_{0}^{1,p}(\Omega)),\
m=1,2,\dots,M\right\\}$

and

$\widehat{X}_{k}^{r}=\left\\{\varphi_{k}\in
L^{2}(I\times\Omega):\varphi_{k}|_{I_{m}}\in\mathbb{P}_{r}(I_{m};W_{0}^{1,p^{\prime}}(\Omega)),\
m=1,2,\dots,M\right\\},$

for some $1<p<\frac{N}{N-1}$ and $p^{\prime}>N$ the dual exponent satisfying
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. In this setting, the embedding
$W_{0}^{1,p^{\prime}}(\Omega)\hookrightarrow C_{0}(\Omega)$, yields that
$\langle v_{0},\varphi_{k,0}^{+}\rangle$ is well defined for all
$\varphi_{k}\in\widehat{X}_{k}^{r}$. With these spaces, the dG($r$)
semidiscrete (in time) solution $v_{k}$ of (1) for
$v_{0}\in\mathcal{M}(\Omega)$ is given by $v_{k}\in\widetilde{X}_{k}^{r}$ that
satisfies

$B(v_{k},\varphi_{k})=\left\langle
v_{0},\varphi_{k,0}^{+}\right\rangle\quad\text{for all
}\;\varphi_{k}\in\widehat{X}_{k}^{r}.$ (9)

Here the bilinear form $B(\cdot,\cdot)$ is defined by

$B(w,\varphi)=\sum_{m=1}^{M}\langle
w_{t},\varphi\rangle_{I_{m}\times\Omega}+({\rm NewA}w,{\rm
NewA}\varphi)_{I\times\Omega}+\sum_{m=2}^{M}([w]_{m-1},\varphi_{m-1}^{+})_{\Omega}+(w_{0}^{+},\varphi_{0}^{+})_{\Omega},$
(10)

where $\langle\cdot,\cdot\rangle_{I_{m}\times\Omega}$ is the duality product
between $L^{2}(I_{m};W^{-1,p^{\prime}}(\Omega))$ and
$L^{2}(I_{m};W^{1,p}_{0}(\Omega))$. In the above definition we use the usual
notation for functions with possible discontinuities at the nodes $t_{m}$:

$w^{+}_{m}=\lim_{\varepsilon\to 0^{+}}w(t_{m}+\varepsilon),\quad
w^{-}_{m}=\lim_{\varepsilon\to
0^{+}}w(t_{m}-\varepsilon),\quad[w]_{m}=w^{+}_{m}-w^{-}_{m}.$ (11)

Due to the fact that $-\Delta:W^{1,p}_{0}(\Omega)\to
W^{-1,p^{\prime}}(\Omega)$ is an isomorphism, see [6, Corollary 2.7] and the
semidiscrete dG method being well defined in the $H^{1}_{0}(\Omega)$ setting,
see [18, p. 205], there exists a unique solution
$v_{k}\in\widetilde{X}_{k}^{r}$ solving (9).

######  Remark 3.1.

Note that whenever $v_{0}\in L^{2}(\Omega)$ the formulation (9) is equivalent
to searching $v_{k}\in X_{k}^{r}$, satisfying

$B(v_{k},\varphi_{k})=\left\langle
v_{0},\varphi_{k,0}^{+}\right\rangle\quad\text{for all }\;\varphi_{k}\in
X_{k}^{r}.$ (12)

Moreover, in the definitions of $\widetilde{X}_{k}^{r},\widehat{X}_{k}^{r}$ we
could equivalently use $H^{1}_{0}(\Omega)$ for test and trial functions on the
intervals $m=2,...,M$, as it was done in [14], since by definition
$v_{k,1}^{-}\in W^{1,p^{\prime}}_{0}(\Omega)\hookrightarrow H^{-1}(\Omega)$.

######  Remark 3.2.

Since we are dealing with homogeneous parabolic problem with constant
coefficients, the discontinuous Galerkin method actually coincides with
subdiagonal Padé approximations and one can use, for example, a rational
representation of the semidiscrete solution. However, it is more convenient
for our analysis to use the definition based on the bilinear form
$B(\cdot,\cdot)$.

Rearranging the terms in (10), we obtain an equivalent (dual) expression for
$B$:

$B(w,\varphi)=-\sum_{m=1}^{M}\langle
w,\varphi_{t}\rangle_{I_{m}\times\Omega}+({\rm NewA}w,{\rm
NewA}\varphi)_{I\times\Omega}-\sum_{m=1}^{M-1}(w_{m}^{-},[\varphi]_{m})_{\Omega}+(w_{M}^{-},\varphi_{M}^{-})_{\Omega}.$
(13)

Notice that for the very weak solution $v$ to (4) and the semidiscrete
solution $v_{k}\in\widetilde{X}_{k}^{r}$ to (9) we have the following
orthogonality property:

$B(v-v_{k},\varphi_{k})=0\quad\text{for all
}\;\varphi_{k}\in\widehat{X}_{k}^{r}.$ (14)

Next we define the fully discrete approximation scheme.

###  3.2. Space discretization

For some $h_{0}>0$ and $h\in(0,h_{0}]$ let $\mathcal{T}$ denote a quasi-
uniform triangulation of $\Omega$ with mesh size $h$, i.e.
$\mathcal{T}=\\{\tau\\}$ is a partition of $\Omega$ into cells (triangles or
tetrahedrons) $\tau$ of diameter $h_{\tau}$ and measure $|\tau|$ such that for
$h=\max_{\tau}h_{\tau}$,

$h_{\tau}\leq h\leq C|\tau|^{\frac{1}{N}},\quad\text{for all
}\;\tau\in\mathcal{T},$

hold. Let $V^{s}_{h}\subset H^{1}_{0}(\Omega)$ be the usual space of
conforming piecewise polynomial finite elements of degree $s$. We define the
following three operators to be used in the sequel: the discrete Laplacian
$\Delta_{h}\colon V^{s}_{h}\to V^{s}_{h}$, defined by

$(-\Delta_{h}v_{h},w_{h})_{\Omega}=(\nabla v_{h},\nabla
w_{h})_{\Omega}\quad\text{for all }\;v_{h},w_{h}\in V^{s}_{h},$ (15)

the $L^{2}$ projection $P_{h}\colon L^{2}(\Omega)\to V^{s}_{h}$, defined by

$(P_{h}v,w_{h})_{\Omega}=(v,w_{h})_{\Omega}\quad\text{for all }\;w_{h}\in
V^{s}_{h},$

and the Ritz projection $R_{h}\colon H^{1}_{0}(\Omega)\to V^{s}_{h}$, defined
by

$(\nabla R_{h}v,\nabla w_{h})_{\Omega}=(\nabla v,\nabla
w_{h})_{\Omega}\quad\text{for all }\;w_{h}\in V^{s}_{h}.$

To obtain the fully discrete approximation of (1) we consider the space-time
finite element space

$X^{r,s}_{k,h}=\Set{v_{kh}\in
X_{k}^{r}}{v_{kh}|_{I_{m}}\in\mathbb{P}_{r}(I_{m};V^{s}_{h}),\
m=1,2,\dots,M}.$ (16)

We define a fully discrete cG($s$)dG($r$) approximation $v_{kh}\in
X^{r,s}_{k,h}$ of (1) by

$B(v_{kh},\varphi_{kh})=\left\langle
v_{0},\varphi_{kh,0}^{+}\right\rangle\quad\text{for all }\;\varphi_{kh}\in
X^{r,s}_{k,h}.$ (17)

Similarly to the time semidiscretization, we have the following orthogonality
relation for the very weak solution $v$ to (4) and the fully discrete solution
$v_{kh}\in X^{r,s}_{k,h}$ to (17):

$B(v-v_{kh},\varphi_{kh})=0\quad\text{for all }\;\varphi_{kh}\in
X^{r,s}_{k,h}.$ (18)

Existence of a unique solution $v_{kh}$ is shown e.g. in [18]. At the end of
this section, we would like to introduce the following truncation argument,
which we will use often in our proofs. For $w_{k},\varphi_{k}\in X_{k}^{r}$,
we let $\tilde{w}_{k}=\chi_{(t_{\tilde{m}},T]}w_{k}$ and
$\tilde{\varphi}_{k}=\chi_{(t_{\tilde{m}},T]}\varphi_{k}$, where
$\chi_{(t_{\tilde{m}},T]}$ is the characteristic function on the interval
$(t_{\tilde{m}},T]$, for some $1\leq\tilde{m}\leq M$, i.e. $\tilde{w}_{k}=0$
on $I_{1}\cup\cdots\cup I_{\tilde{m}}$ for some $\tilde{m}$ and
$\tilde{w}_{k}={w}_{k}$ on the remaining time intervals. Then from (10), we
have the identity

$B(\tilde{w}_{k},\varphi_{k})=B(w_{k},\tilde{\varphi}_{k})+(w_{k,\tilde{m}}^{-},\varphi_{k,\tilde{m}}^{+})_{\Omega}.$
(19)

The same identity holds of course for fully discrete functions
$w_{kh},\varphi_{kh}\in X^{r,s}_{k,h}$.

##  4\. Parabolic smoothing

In this section we review and establish smoothing properties of the continuous
and discrete solutions, which are essential for the establishment of our main
results.

###  4.1. Smoothing estimates for the continuous problem

It is well known that homogeneous parabolic problems have a strong smoothing
effect. In particular, for $v_{0}\in L^{2}(\Omega)$, the solution $v$ to the
problem (1) has the following smoothing property, see [3, Chapter 1, Eq. 1.14]

$\|\partial_{t}^{l}v(t)\|_{L^{2}(\Omega)}+\|(-\Delta)^{l}v(t)\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l}}\|v_{0}\|_{L^{2}(\Omega)}\quad
t>0,\quad l=0,1,\dots.$ (20)

######  Remark 4.1.

In many situations it is sufficient to have smoothing type estimates in
$L^{2}$ norms and the corresponding smoothing results, for example in $L^{p}$
norms, can obtained by the Gagliardo-Nirenberg inequality

$\left\lVert g\right\rVert_{L^{p}(B)}\leq C\left\lVert
g\right\rVert^{\alpha}_{H^{2}(B)}\left\lVert
g\right\rVert^{1-\alpha}_{L^{2}(B)},\quad 2\leq p\leq\infty,\quad\text{for}\
\alpha=\frac{N}{4}-\frac{N}{2p},$ (21)

which holds for any subdomain $B\subset\Omega$ fulfilling the cone condition
(in particular for $B=\Omega$) and for all $g\in H^{2}(B)$, see [1, Theorem
3]. In particular, for $p=\infty$ on convex domains, we have

$\left\lVert g\right\rVert_{L^{\infty}(\Omega)}\leq C\left\lVert\Delta
g\right\rVert^{\frac{N}{4}}_{L^{2}(\Omega)}\left\lVert
g\right\rVert^{1-\frac{N}{4}}_{L^{2}(\Omega)},\quad\text{for}\ g\in
H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$ (22)

Thus using (22) and the smoothing estimates (20) for $v_{0}\in L^{2}(\Omega)$,
we immediately obtain

$\|v(t)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t^{\frac{N}{4}}}\|v_{0}\|_{L^{2}(\Omega)}\quad
t>0.$ (23)

Using a duality argument and the smoothing estimates, this result can be
easily extended to the case $v_{0}\in\mathcal{M}(\Omega)$. First we derive the
explicit time dependence of the constant occuring in the estimate of Theorem
2.1.

######  Lemma 4.2.

Let $v_{0}\in\mathcal{M}(\Omega)$ and $v$ be the very weak solution of (1).
Then

$\|v(t)\|_{L^{2}(\Omega)}\leq\frac{C}{t^{\frac{N}{4}}}\|v_{0}\|_{\mathcal{M}(\Omega)}\quad
t>0.$ (24)

######  Proof 4.3.

We will establish the result for $t=T$. Define $y$ to be the solution to the
dual problem

$\left\\{\begin{aligned} -\partial_{t}y-\Delta y&=0&&\text{in
}I\times\Omega\\\ y&=0&&\text{on }I\times\partial\Omega\\\
y(T)&=v(T)&&\text{in }\Omega.\end{aligned}\right.$ (25)

Then (23) applied to $y$ yields $y(0)\in C_{0}(\Omega)$ and we have the
estimate

$\displaystyle\|v(T)\|_{L^{2}(\Omega)}^{2}=(v(T),y(T))_{\Omega}=\langle
v_{0},y(0)\rangle$ $\displaystyle\leq C\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}\|y(0)\|_{L^{\infty}(\Omega)}\leq
CT^{-\frac{N}{4}}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}\|v(T)\|_{L^{2}(\Omega)}.$ (26)

Canceling $\|v(T)\|_{L^{2}(\Omega)}$ gives the result.

Let $v_{0}\in\mathcal{M}(\Omega)$ and $v$ be the very weak solution of (1).
Then

$\|\partial_{t}^{l}v(t)\|_{L^{2}(\Omega)}+\|(-\Delta)^{l}v(t)\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l+\frac{N}{4}}}\|v_{0}\|_{\mathcal{M}(\Omega)}\quad
t>0,\quad l=0,1,\dots.$ (27)

######  Proof 4.4.

This is a direct consequence of Theorem 2.1, Lemma 2.2 and the above smoothing
result. The time dependency of the constant can be seen, by fixing $t\in(0,T)$
and setting $\tau=\frac{t}{2}$ in Lemma 2.2. Then by Lemma 4.2, we have
$\left\lVert v\left(\frac{t}{2}\right)\right\rVert_{L^{2}(\Omega)}\leq
C\left(\frac{2}{t}\right)^{\frac{N}{4}}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}$. By (20), it also holds

$\|\partial_{t}^{l}v(t)\|_{L^{2}(\Omega)}+\|(-\Delta)^{l}v(t)\|_{L^{2}(\Omega)}\leq\frac{C}{(t-\frac{t}{2})^{l}}\left\lVert
v\left(\frac{t}{2}\right)\right\rVert_{L^{2}(\Omega)}\leq\frac{C}{t^{l+\frac{N}{4}}}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}.$ (28)

By applying the Gagliardo-Nirenberg inequality (22), we immediately obtain:
Let $v_{0}\in\mathcal{M}(\Omega)$ and $v$ be the very weak solution of (1).
Then

$\|v(t)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t^{\frac{N}{2}}}\|v_{0}\|_{\mathcal{M}(\Omega)},\quad
t>0.$ (29)

###  4.2. Smoothing estimates for the discrete solutions

For the time discontinuous Galerkin solutions, both the semidiscrete and the
fully discrete, similar smoothing type estimates also hold (see Theorems
3,4,5,10 in [12] for general $L^{p}$ norms, cf. also [7, Theorem 5.1] for the
case of the $L^{2}$ norm).

######  Lemma 4.5.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X^{r,s}_{k,h}$ be the semidiscrete and
fully discrete solutions of (9) and (17), respectively, with $v_{0}\in
L^{2}(\Omega)$. Then, there exists a constant $C$ independent of $k$ and $h$
such that

$\displaystyle\sup_{t\in
I_{m}}\|\partial_{t}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in I_{m}}\|\Delta
v_{k}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[v_{k}]_{m-1}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\frac{C}{t_{m}}\|v_{0}\|_{L^{2}(\Omega)},$
$\displaystyle\sup_{t\in
I_{m}}\|\partial_{t}v_{kh}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|\Delta_{h}v_{kh}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[v_{kh}]_{m-1}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\frac{C}{t_{m}}\|v_{0}\|_{L^{2}(\Omega)},$

for $m=1,2,\dots,M$. For $m=1$ the jump term is understood as
$[v_{k}]_{0}=v_{k,0}^{+}-v_{0}$ and $[v_{kh}]_{0}=v_{kh,0}^{+}-P_{h}v_{0}$.

The above estimates immediately imply the following stability result. Under
the assumptions of Lemma 4.5, we have

$\sum_{m=1}^{M}\left(\|\partial_{t}v_{k}\|_{L^{1}(I_{m};L^{2}(\Omega))}+\|\Delta
v_{k}\|_{L^{1}(I_{m};L^{2}(\Omega))}+k_{m}\|\Delta
v_{k,m}^{+}\|_{L^{2}(\Omega)}+\|[v_{k}]_{m-1}\|_{L^{2}(\Omega)}\right)\leq
C\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}$

and

$\sum_{m=1}^{M}\Bigl{(}\|\partial_{t}v_{kh}\|_{L^{1}(I_{m};L^{2}(\Omega))}+\|\Delta_{h}v_{kh}\|_{L^{1}(I_{m};L^{2}(\Omega))}+k_{m}\|\Delta_{h}v_{kh,m}^{+}\|_{L^{2}(\Omega)}+\|[v_{kh}]_{m-1}\|_{L^{p}(\Omega)}\Bigr{)}\leq
C\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}.$

For sufficiently many time steps, applying Lemma 4.5 iteratively, we have the
following result.

######  Lemma 4.6.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X^{r,s}_{k,h}$ be the semidiscrete and
fully discrete solutions of (9) and (17), respectively, with $v_{0}\in
L^{2}(\Omega)$. Then, for any $m\in\\{1,2,\dots M\\}$, any $l\leq m$, there
hold

$\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta)^{l-1}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta)^{l}v_{k}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[(-\Delta)^{l-1}v_{k}]_{m-1}\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l}_{m}}\|v_{0}\|_{L^{2}(\Omega)}$

and

$\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta_{h})^{l-1}v_{kh}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta_{h})^{l}v_{kh}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[(-\Delta_{h})^{l-1}v_{kh}]_{m-1}\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l}_{m}}\|v_{0}\|_{L^{2}(\Omega)},$

provided $k\leq\frac{t_{m}}{l+1}$.

Using the continuous (22) and the discrete version of the Gagliardo-Nirenberg
inequality, namely

$\|\chi\|_{L^{\infty}(\Omega)}\leq
C\|\Delta_{h}\chi\|^{\frac{N}{4}}_{L^{2}(\Omega)}\|\chi\|^{1-\frac{N}{4}}_{L^{2}(\Omega)},\quad\text{for
all }\;\chi\in V_{h}^{s},$ (30)

which for example was established for smooth domains in [9, Lemma 3.3], but
the proof is valid for convex domains as well, we immediately obtain the
following smoothing result. Under the assumption of Lemma 4.5 for all
$m=2,3,\dots,M$, we have

$\sup_{t\in
I_{m}}\|v_{k}(t)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t_{m}^{N/4}}\|v_{0}\|_{L^{2}(\Omega)}\quad\text{and}\quad\sup_{t\in
I_{m}}\|v_{kh}(t)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t_{m}^{N/4}}\|v_{0}\|_{L^{2}(\Omega)}.$

Similarly to the continuous case, using a duality argument, the above
smoothing results can be extended to $v_{0}\in\mathcal{M}(\Omega)$.

######  Lemma 4.7.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X^{r,s}_{k,h}$ be the semidiscrete and
the fully discrete solutions of (9) and (17) respectively. For any
$m\in\\{1,2,\dots M\\}$, there hold

$\|v_{k}(t_{m})\|_{L^{2}(\Omega)}+\|v_{kh}(t_{m})\|_{L^{2}(\Omega)}\leq\frac{C}{t^{\frac{N}{4}}_{m}}\|v_{0}\|_{\mathcal{M}(\Omega)}.$
(31)

######  Proof 4.8.

It is sufficient to establish the result only at node points $t_{m}$,
$m=2,3,\dots,M$. Define $y_{k}\in\widehat{X}_{k}^{r}$ to be the semidiscrete
solution of the backward problem

$B(\psi_{k},y_{k})=(\psi_{k,m}^{-},v_{k}(t_{m}))_{\Omega},\quad\forall\psi_{k}\in\widetilde{X}_{k}^{r}.$

Choosing $\psi_{k}=v_{k}$ and using Corollary 4.2 for the backward problem, we
have

$\|v_{k}(t_{m})\|^{2}_{L^{2}(\Omega)}=B(v_{k},y_{k})=\left\langle
v_{0},y_{k,0}^{+}\right\rangle\leq\|v_{0}\|_{\mathcal{M}(\Omega)}\|y_{k}(0)\|_{L^{\infty}(\Omega)}\leq\frac{C}{t_{m}^{N/4}}\|v_{0}\|_{\mathcal{M}(\Omega)}\|v_{k}(t_{m})\|_{L^{2}(\Omega)}.$

Canceling, we obtain the result for the time semidiscrete solution $v_{k}$.
The argument for the fully discrete solution $v_{kh}$ is almost identical.

From Lemma 4.6, we can obtain the following result

######  Lemma 4.9.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X^{r,s}_{k,h}$ be the semidiscrete and
the fully discrete solutions of (9) and (17) respectively. Let
$m\in\\{1,2,\dots M\\}$ large enough and $l\leq m$, such that
$k\leq\min\\{\frac{t_{m}}{4},\frac{t_{m}}{2(l+1)}\\}$, then there hold

$\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta)^{l-1}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta)^{l}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}k_{m}^{-1}\|[(-\Delta)^{l-1}v_{k}]_{m-1}\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l+\frac{N}{4}}_{m}}\|v_{0}\|_{\mathcal{M}(\Omega)}$
(32)

and

$\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta_{h})^{l-1}v_{kh}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta_{h})^{l}v_{kh}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}k_{m}^{-1}\|[(-\Delta_{h})^{l-1}v_{kh}]_{m-1}\|_{L^{2}(\Omega)}\leq\frac{C}{t^{l+{\frac{N}{4}}}_{m}}\|v_{0}\|_{\mathcal{M}(\Omega)}.$
(33)

######  Proof 4.10.

We will only establish semidiscrete smoothing estimates for measure valued
initial data, the analysis for the fully discrete solution is similar.
Combining Lemma 4.6 with Lemma 4.7, gives us for all $m>\tilde{m}+l$:

$\displaystyle\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta)^{l-1}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta)^{l}v_{k}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[(-\Delta)^{l-1}v_{k}]_{m-1}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{C}{(t_{m}-t_{\tilde{m}})^{l}}\|v_{k}(t_{\tilde{m}})\|_{L^{2}\Omega)}\leq\frac{C}{(t_{m}-t_{\tilde{m}})^{l}\
t_{\tilde{m}}^{\frac{N}{4}}}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}.$

For fixed $t_{m}$ with $m$ large enough such that
$k\leq\min\\{\frac{t_{m}}{4},\frac{t_{m}}{2(l+1)}\\}$ we apply the above
argument to $t_{\tilde{m}}$ such that
$\frac{t_{m}}{2}\in(t_{\tilde{m}-1},t_{\tilde{m}}]$. By the requirements on
$k$ we obtain on the one hand that there are at least $l$ timesteps between
$t_{\tilde{m}}$ and $t_{m}$, which allows us to use the smoothing estimate of
Lemma 4.6 for the specified $l$. On the other hand, we obtain

$t_{m}-t_{\tilde{m}}\geq\frac{t_{m}}{2}-k\geq\frac{t_{m}}{2}-\frac{t_{m}}{4}=\frac{t_{m}}{4}\qquad\text{yielding}\qquad(t_{m}-t_{\tilde{m}})^{-l}\leq
4^{l}\frac{1}{t_{m}^{l}}.$

The choice $\frac{t_{m}}{2}\leq t_{\tilde{m}}$ gives
$t_{\tilde{m}}^{-\frac{N}{4}}\leq 2^{\frac{N}{4}}t_{m}^{-\frac{N}{4}}$, which
allows us to eliminate $t_{\tilde{m}}$ in the final bound and obtain

$\sup_{t\in
I_{m}}\|\partial_{t}(-\Delta)^{l-1}v_{k}(t)\|_{L^{2}(\Omega)}+\sup_{t\in
I_{m}}\|(-\Delta)^{l}v_{k}(t)\|_{L^{2}(\Omega)}+k_{m}^{-1}\|[(-\Delta)^{l-1}v_{k}]_{m-1}\|_{L^{2}(\Omega)}\leq
C(l,N)t_{m}^{-l-\frac{N}{4}}\left\lVert
v_{0}\right\rVert_{\mathcal{M}(\Omega)}.$ (34)

##  5\. Smoothing type error estimates

First we review smoothing results with the initial data in $L^{2}(\Omega)$ and
then extend the corresponding results to ${\mathcal{M}(\Omega)}$.

###  5.1. Review of pointwise smoothing error estimates for $v_{0}\in
L^{2}(\Omega)$

In [14], we have established the following pointwise fully discrete error
estimate.

######  Proposition 1.

Let $v_{0}\in L^{2}(\Omega)$, let $v$ and $v_{kh}\in X_{k,h}^{r,1}$ satisfy
(1) and (17), respectively. Then for any subdomain $\Omega_{0}$ with
$\overline{\Omega}_{0}\subset\Omega$ there holds

$\|(v-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,\Omega_{0})\left(\ell_{kh}h^{2}+k^{2r+1}\right)\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)},$

where $\ell_{kh}=\ln{\frac{T}{k}}+\lvert\ln{h}\rvert$ and $C(T,\Omega_{0})$ is
a constant that depends on $T$ and $\Omega_{0}$ and the explicit form can be
traced from the proof.

The proof of the above result was based on the following splitting of the
error

$(v-v_{kh})(T)=(v-v_{k})(T)+(R_{h}v_{k}-v_{kh})(T)+(v_{k}-R_{h}v_{k})(T).$
(35)

Then each term was treated separately. The first error term was estimated in
[14, Theorem 3.8] by

$\lVert(v-v_{k})(T)\rVert_{L^{\infty}(\Omega)}\leq C(T)k^{2r+1}\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)},$ (36)

with $C(T)\sim T^{-(2r+1+\frac{N}{4})}$. The above estimate follows from (see
[14, Lem. 7.2])

$\left\lVert(-\Delta)^{j}(v-v_{k})(T)\right\rVert_{L^{2}(\Omega)}\leq
C_{j}(T)k^{2r+1}\left\lVert v_{0}\right\rVert_{L^{2}(\Omega)},\quad
j=0,1,\dots.$ (37)

The second error term in (35) satisfies,

$\|(R_{h}v_{k}-v_{kh})(T)\|_{L^{\infty}(\Omega)}\leq
C(T)\ln{\frac{T}{k}}h^{2}\left\lVert v_{0}\right\rVert_{L^{2}(\Omega)},$ (38)

which followed from (see [14, Lem. 8.2–8.3])

$\|(-\Delta_{h})^{j}(R_{h}v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)}\leq
C_{j}(T)\ln{\frac{T}{k}}h^{2}\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)},\quad j=0,1,$ (39)

and the discrete Gagliardo-Nirenberg inequality (30). Here, we point out that
the treatment of the first and the second terms of (35) do not require the
condition $\overline{\Omega}_{0}\subset\Omega$, they are global in nature.
Finally, the estimate of the last term in (35) follows from the interior
elliptic error estimate (cf. [17])

$\|(v_{k}-R_{h}v_{k})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,\Omega_{0})\lvert\ln{h}\rvert h^{2}\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)}.$ (40)

###  5.2. Pointwise smoothing error estimates for
$v_{0}\in\mathcal{M}(\Omega)$

We now turn towards proving the pointwise error estimate for measure valued
initial data. To this end, first recall that in [14, Lemma 5.1] we have shown
the following $L^{2}$ error estimate for parabolic problems with initial data
in $\mathcal{M}(\Omega)$, where for the spatial estimate we impose a condition
on the support of $v_{0}$.

######  Lemma 5.1.

Let $v_{0}\in\mathcal{M}(\Omega)$ with
$\operatorname{supp}v_{0}\subset\Omega_{0}$ for some subdomain
$\Omega_{0}\subset\overline{\Omega}_{0}\subset\Omega$ and let $v$,
$v_{k}\in\widetilde{X}_{k}^{r}$ and $v_{kh}\in X_{k,h}^{r,1}$ the continuous,
semidiscrete and fully discrete solutions to (1), (9) and (17) respectively.
Then there hold the estimates

$\displaystyle\|(v-v_{k})(T)\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C(T)k^{2r+1}\|v_{0}\|_{\mathcal{M}(\Omega)}$
$\displaystyle\|(v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)}$ $\displaystyle\leq
C(\Omega_{0},T)\ell_{kh}h^{2}\|v_{0}\|_{\mathcal{M}(\Omega)},$

where $\ell_{kh}=\ln{\frac{T}{k}}+\lvert\ln{h}\rvert$ and $C(T,\Omega_{0})$ is
a constant that depends on $T$ and $\Omega_{0}$ and the explicit form can be
traced from the proof.

Our main result can now be obtained directly by introducing an auxiliary
solution and the smoothing results presented in Section 4. We first prove the
error estimate for the spatial discretization. The proof of the error estimate
for the time semidiscretization follows the same steps under milder
assumptions, see Lemma 5.4 below.

######  Theorem 5.2.

Let $v_{0}\in\mathcal{M}(\Omega)$, let $v_{k}\in\widetilde{X}_{k}^{r}$ and
$v_{kh}\in X_{k,h}^{r,1}$ satisfy (9) and (17), respectively. Then for any
subdomain $\Omega_{0}$ with $\overline{\Omega}_{0}\subset\Omega$ and
$\operatorname{supp}v_{0}\subset\Omega_{0}$ there holds

$\|(v_{k}-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,\Omega_{0})\ell_{kh}h^{2}\|v_{0}\|_{\mathcal{M}(\Omega)},$

where $\ell_{kh}=\ln{\frac{T}{k}}+\lvert\ln{h}\rvert$ and $C(T,\Omega_{0})$ is
a constant that depends on $T$ and $\Omega_{0}$ and the explicit form can be
traced from the proof.

######  Proof 5.3.

As done in the proofs of the smoothing results, we begin by splitting the time
interval. To this end let $\tilde{m}$ be such that $\frac{T}{2}\in
I_{\tilde{m}}$. We introduce a fully discrete auxiliary state $\hat{v}_{kh}\in
X_{k,h}^{r,1}$, defined by

$B(\hat{v}_{kh},\varphi_{kh})=(v_{k,\tilde{m}-1}^{-},\varphi_{kh,\tilde{m}-1}^{+})_{\Omega}\quad\text{for
all}\quad\varphi_{kh}\in X_{k,h}^{r,1}.$

Note that by definition $\hat{v}_{kh}\equiv 0$ on $I_{1}\cup...\cup
I_{\tilde{m}-1}$ and it satisfies a discrete problem on
$I_{\tilde{m}}\cup...\cup I_{M}$ with initial condition
$v_{k,\tilde{m}-1}^{-}$ at time $t_{\tilde{m}-1}$. By the triangle inequality,
we obtain

$\|(v_{k}-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq\|(v_{k}-\hat{v}_{kh})(T)\|_{L^{\infty}(\Omega_{0})}+\|(\hat{v}_{kh}-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})},$
(41)

where for the first term, we obtain with Proposition 1 and the semidiscrete
parabolic smoothing result of Lemma 4.7

$\displaystyle\|(v_{k}-\hat{v}_{kh})(T)\|_{L^{\infty}(\Omega_{0})}$
$\displaystyle\leq
C(T-t_{\tilde{m}-1},\Omega_{0})\ell_{kh}h^{2}\|v_{k,\tilde{m}-1}^{-}\|_{L^{2}(\Omega)}$
(42) $\displaystyle\leq
C(T-t_{\tilde{m}-1},\Omega_{0})t_{\tilde{m}-1}^{-\frac{N}{4}}\ell_{kh}h^{2}\|v_{0}\|_{\mathcal{M}(\Omega)}.$
(43)

For the second error term, we observe that the difference
$\hat{v}_{kh}-v_{kh}$ satisfies a fully discrete parabolic equation on the
intervals $I_{\tilde{m}}\cup...\cup I_{M}$ for the initial data
$v_{k,\tilde{m}-1}^{-}-v_{kh,\tilde{m}-1}^{-}$. Hence, the discrete Gagliardo-
Nirenberg inequality (30) and the fully discrete smoothing results of Lemmas
4.5 and 4.6 yield

$\displaystyle\|(\hat{v}_{kh}-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}$
$\displaystyle\leq
C\|(\hat{v}_{kh}-v_{kh})(T)\|_{L^{2}(\Omega)}^{\frac{1}{2}}\|\Delta_{h}(\hat{v}_{kh}-v_{kh})(T)\|_{L^{2}(\Omega)}^{\frac{1}{2}}$
$\displaystyle\leq
C(T-t_{\tilde{m}-1})^{-\frac{1}{2}-\frac{N}{4}}\|v_{k,\tilde{m}-1}^{-}-v_{kh,\tilde{m}-1}^{-}\|_{L^{2}(\Omega)}$

We apply Lemma 5.1 in order to estimate the $L^{2}$-error of the full
discretization at the intermediate point in time, which yields

$\|(\hat{v}_{kh}-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(t_{\tilde{m}-1},\Omega_{0})(T-t_{\tilde{m}-1})^{-\frac{1}{2}-\frac{N}{4}}\ell_{kh}h^{2}\|v_{0}\|_{\mathcal{M}(\Omega)}.$
(44)

Since the assumptions on $k$ and $\tilde{m}$ yield $\frac{T}{4}\leq
t_{\tilde{m}-1}\leq\frac{T}{2}$ and $\frac{T}{2}\leq
T-t_{\tilde{m}-1}\leq\frac{3T}{4}$, as before we can replace all quantities
involving $t_{\tilde{m}-1}$ by ones only dependent of $T$, which concludes the
proof.

Note that by exactly the same technique, we can also derive the corresponding
error estimate for the semidiscrete problem, which is global in $\Omega$ and
no constraint on $\operatorname{supp}v_{0}$ is required. This is due to the
fact, that the semidiscrete results of (36) and [14, Lemma 5.1] hold in this
more general setting. There holds the following result.

######  Lemma 5.4.

Let $v_{0}\in\mathcal{M}(\Omega)$ and $v$ and $v_{k}\in\widetilde{X}_{k}^{r}$
be the very weak and semidiscrete solutions to (4) and (9) respectively. Then
there holds

$\lVert(v-v_{k})(T)\rVert_{L^{\infty}(\Omega)}\leq
C(T)k^{2r+1}\|v_{0}\|_{\mathcal{M}(\Omega)},$

with $C(T)\sim T^{-(2r+1+\frac{N}{2})}$.

##  6\. Higher order space discretizations

Our main result from the previous section, Theorem 5.2, was established for
piecewise linear finite elements only and does not require any additional
smoothness assumptions on the solutions beyond $H^{2}$ regularity that is
provided by the convexity of the domain. If additional regularity is
available, for example,

$|v|_{H^{3}(\Omega)}\leq C\|\Delta v\|_{H^{1}_{0}(\Omega)}$ (45)

for any $v\in H^{1}_{0}(\Omega)$ with $\Delta v\in H^{1}_{0}(\Omega)$, then
the results of Proposition 1 can be extended (with an improved rate) to the
case of quadratic Lagrange finite elements which we will denote by $V_{h}^{2}$
in this section.

######  Remark 6.1.

Since due to Remark 3.1 for each $t\in(t_{1},T]$, the solution $v_{k}$ to the
semidiscrete problem (9) satisfies $v_{k}(t)\in H^{1}_{0}(\Omega)$, one can
also show straightforwardly that

$\displaystyle\Delta v_{k}(t),\ \partial_{t}\Delta v_{k}(t)\in
H^{1}_{0}(\Omega)\ \text{ for all }t\in I_{m},\ m\geq
2\quad\text{and}\quad\Delta^{2}v_{k}(t)\in H^{1}_{0}(\Omega)\ \text{ for all
}t\in I_{m},\ m\geq 3.$

Additional regularity is available on special domains, for example on
rectangles, right or equilateral triangles. We make the following assumption
of the domain $\Omega$.

######  Assumption 1.

For every $u\in H^{1}_{0}(\Omega)$ with $\Delta u\in H^{1}_{0}(\Omega)$ there
holds $u\in H^{3}(\Omega)$. Moreover, there exists a constant $C$ independent
of $u$ such that

$\|u\|_{H^{3}(\Omega)}\leq C\|{\rm NewA}\Delta u\|_{L^{2}(\Omega)}.$ (46)

######  Example 6.2.

This assumption holds for example on a rectangle, see [10, Lemma 2.4]. In this
case the solution $u$ to the elliptic equation

$\displaystyle-\Delta u$ $\displaystyle=f\quad\text{in}\ \Omega$
$\displaystyle u$ $\displaystyle=0\quad\text{on}\ \partial\Omega,$

with $f\in H^{1}_{0}(\Omega)$ possesses the $H^{3}(\Omega)$ regularity with
the estimate

$\|u\|_{H^{3}(\Omega)}\leq C\|{\rm NewA}f\|_{L^{2}(\Omega)}$

holds. Thus, Assumption 1 is satisfied in this case.

######  Lemma 6.3.

Let $\Omega$ satisfy Assumption 1.

  1. (1)

Let $u,\Delta u\in H^{1}_{0}(\Omega)$, and $\Delta^{2}u\in L^{2}(\Omega)$.
Then there holds

$\|u\|^{2}_{H^{3}(\Omega)}\leq C\|\Delta
u\|_{L^{2}(\Omega)}\|\Delta^{2}u\|_{L^{2}(\Omega)}.$ (47)

  2. (2)

Let $\Omega_{0}$ be a subdomain with $\overline{\Omega}_{0}\subset\Omega$, let
$u,\Delta u,\Delta^{2}u\in H^{1}_{0}(\Omega)$, and $\Delta^{3}u\in
L^{2}(\Omega)$. Then there holds

$\|u\|^{2}_{H^{5}(\Omega_{0})}\leq
C\|\Delta^{2}u\|_{L^{2}(\Omega)}\|\Delta^{3}u\|_{L^{2}(\Omega)}.$ (48)

######  Proof 6.4.

  1. (1)

For $v\in H^{1}_{0}(\Omega)$ with $\Delta v\in L^{2}(\Omega)$ one directly
obtains

$\|\nabla v\|^{2}_{L^{2}(\Omega)}\leq\|v\|_{L^{2}(\Omega)}\|\Delta
v\|_{L^{2}(\Omega)}.$

Due to $\Delta u\in H^{1}_{0}(\Omega)$ and $\Delta^{2}u\in L^{2}(\Omega)$ this
inequality can be applied to $v=\Delta u$ leading to

$\|\nabla\Delta u\|^{2}_{L^{2}(\Omega)}\leq\|\Delta
u\|_{L^{2}(\Omega)}\|\Delta^{2}u\|_{L^{2}(\Omega)}.$

Thus, Assumption 1 implies the desired estimate.

  2. (2)

Using a higher interior regularity result, see [8, Chapter 6.3, Theorem 2], we
obtain

$\|u\|_{H^{5}(\Omega_{0})}\leq C(\|\Delta
u\|_{H^{3}(\Omega_{0})}+\|u\|_{L^{2}(\Omega)}).$

Since $\Delta^{2}u\in H^{1}_{0}(\Omega)$ and $\Delta^{3}u\in L^{2}(\Omega)$ we
can apply (47) to $\Delta u$ leading to

$\|\Delta u\|^{2}_{H^{3}(\Omega_{1})}\leq\|\Delta u\|^{2}_{H^{3}(\Omega)}\leq
C\|\Delta^{2}u\|_{L^{2}(\Omega)}\|\Delta^{3}u\|_{L^{2}(\Omega)}.$

This leads to

$\|u\|^{2}_{H^{5}(\Omega_{0})}\leq
C\|\Delta^{2}u\|_{L^{2}(\Omega)}\|\Delta^{3}u\|_{L^{2}(\Omega)}+C\|\Delta
u\|_{L^{2}(\Omega)}\|\Delta^{2}u\|_{L^{2}(\Omega)},$

which proves the desired result by $\|\Delta^{j}u\|_{L^{2}(\Omega)}\leq
C\|\Delta^{j+1}u\|_{L^{2}(\Omega)}$ for $j=1,2$.

Complementing the standard error estimates for the Ritz projection in the
$L^{2}$ and $H^{1}$ norms, under Assumption 1, we also have the following
negative norm estimate. Note that even though no $H^{3}$ regularity of the
solution $u$ is used explicitly in the estimates, the duality argument used to
prove the result, requires the assumption to hold true for any $H^{1}_{0}$
right hand side.

######  Lemma 6.5.

Let $u\in H^{1}_{0}(\Omega)$ and Assumption 1 hold true. Then it holds

$\|u-R_{h}u\|_{H^{-1}(\Omega)}\leq
Ch^{2}\|\nabla(u-R_{h}u)\|_{L^{2}(\Omega)}.$

If further $u\in H^{2}(\Omega)$, then it holds

$\|u-R_{h}u\|_{H^{-1}(\Omega)}\leq Ch^{3}\|u\|_{H^{2}(\Omega)}\leq
Ch^{3}\|\Delta u\|_{L^{2}(\Omega)}.$

######  Proof 6.6.

The first estimate is proved by a duality argument in [4, Theorem 5.8.3]. The
second estimate then follows with the standard $H^{1}$ error estimate and
$H^{2}$ regularity.

Under Assumption 1 we can establish the main results of this section. We first
consider again the case of $L^{2}$ initial data. The extension to
$v_{0}\in\mathcal{M}(\Omega)$ then follows analogously to the case of linear
finite elements.

######  Theorem 6.7.

Let $v_{0}\in L^{2}(\Omega)$, let $v$ and $v_{kh}\in X_{k,h}^{r,2}$ satisfy
(1) and (17), respectively. Then for any subdomain $\Omega_{0}$ with
$\overline{\Omega}_{0}\subset\Omega$ there holds

$\|(v-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,d)\left(\ell_{k}h^{3}+k^{2r+1}\right)\left\lVert
v_{0}\right\rVert_{L^{2}(\Omega)},$ (49)

where $\ell_{k}=\ln{\frac{T}{k}}$,
$d=\operatorname*{dist}(\Omega_{0},\partial\Omega)$ and $C(T,d)$ is a constant
depending on $T$ and $d$.

###  6.1. Proof of Theorem 6.7

The exact dependence of the constant $C$ on $T$ and $d$ is available in the
proof of this result. The rest of this section is devoted to the establishment
of the above theorem. The proof for the quadratic case is similar to the proof
for the piecewise linear case, but requires some modifications. As it was done
in [14], we split the error as

$(v-v_{kh})(T)=(v-v_{k})(T)+(v_{k}-R_{h}v_{k})(T)+(R_{h}v_{k}-v_{kh})(T):=T_{1}+T_{2}+T_{3}.$
(50)

The first time semidiscrete term $T_{1}$ is already estimated in [14, Theorem
3.8]. The second term $T_{2}$ can again be estimated by the interior pointwise
error estimates of [17, Theorem 5.1],

$\|(v_{k}-R_{h}v_{k})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C\|v_{k}(T)-\chi\|_{L^{\infty}(\Omega_{d})}+Cd^{-N/2}\|(v_{k}-R_{h}v_{k})(T)\|_{L^{2}(\Omega)},$
(51)

for any $\chi\in V^{2}_{h}$, where $\Omega_{d}$ is a subdomain satisfying
$\overline{\Omega}_{0}\subset\Omega_{d}\subset\overline{\Omega}_{d}\subset\Omega$
and $d=\operatorname*{dist}(\Omega_{0},\partial\Omega_{d})$. We note that in
contrast to the linear elements, in the above estimate the logarithmic term is
not needed. By the approximation theory and the Sobolev embedding
$H^{5}(\Omega_{d})\hookrightarrow W^{3,\infty}(\Omega_{d})$ (see e.g. [2,
Theorem 4.12]), Lemma 6.3 and the discrete parabolic smoothing result of Lemma
4.5, we obtain

$\displaystyle\|v_{k}(T)-\chi\|_{L^{\infty}(\Omega_{d})}$ $\displaystyle\leq
Ch^{3}\|v_{k}(T)\|_{W^{3,\infty}(\Omega_{d})}\leq
Ch^{3}\|v_{k}(T)\|_{H^{5}(\Omega_{d})}$ $\displaystyle\leq
Ch^{3}\|\Delta^{2}v_{k}(T)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}\|\Delta^{3}v_{k}(T)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}\leq\frac{Ch^{3}}{T^{\frac{5}{2}}}\|v_{0}\|_{L^{2}(\Omega)}.$

The pollution term $\|(v_{k}-R_{h}v_{k})(T)\|_{L^{2}(\Omega)}$ from (51), can
be estimated using global elliptic estimates in $L^{2}$ norm, Lemma 6.3 and
Lemma 4.9 as

$\|(v_{k}-R_{h}v_{k})(T)\|_{L^{2}(\Omega)}\leq
Ch^{3}\|v_{k}(T)\|_{H^{3}(\Omega)}\leq Ch^{3}\|\Delta
v_{k}(T)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}\|\Delta^{2}v_{k}(T)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}\leq\
\frac{Ch^{3}}{T^{\frac{3}{2}}}\|v_{0}\|_{L^{2}(\Omega)}.$ (52)

Thus,

$\|(v_{k}-R_{h}v_{k})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,\Omega_{0})h^{3}\|v_{0}\|_{L^{2}(\Omega)},$

and it remains to estimate the last term $T_{3}$ of (50). As done in [14,
Lemmas 8.2–8.3], this will be achieved by estimating

$\|(-\Delta_{h})^{j}(R_{h}v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)},\quad j=0,1,$
(53)

and the discrete Gagliardo-Nirenberg inequality (30). The proof of the above
estimates was facilitated by the following technical lemma, see [14, Lemma
8.1].

######  Lemma 6.8.

Let $v_{0}\in L^{2}(\Omega)$, let $v$ and $v_{kh}\in X_{k,h}^{r,1}$ satisfy
(1) and (17), respectively. There exists a constant $C$ independent of $k$,
$h$, and $T$ such that

$\|\Delta_{h}^{-1}(P_{h}v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)}\leq
Ch^{2}\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}.$

In order to prove Theorem 6.7 we thus first extend Lemma 6.8 to quadratic
finite elements in space, in order to estimate the terms of (53).

######  Lemma 6.9.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X_{k,h}^{r,2}$ be the semidiscrete and
fully discrete solutions of (9) and (17), respectively for $v_{0}\in
L^{2}(\Omega)$. Then there exists a constant $C$ independent of $h$,$k$ and
$T$ such that

$\|\Delta_{h}^{-2}\left(P_{h}v_{k}-v_{kh}\right)(T)\|_{L^{2}(\Omega)}\leq
Ch^{3}\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}.$ (54)

######  Proof 6.10.

Let $z_{kh}\in X_{k,h}^{r,2}$ be the solution to a dual problem with
$z_{kh}(T)=\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})(T)$, i.e.

$B(\chi_{kh},z_{kh})=\left(\chi_{kh}(T),\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})(T)\right)\quad\text{for
all}\quad\chi_{kh}\in X_{k,h}^{r,2}.$

Choosing $\chi_{kh}=\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})$, we obtain

$\mathcal{Z}:=\|\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})(T)\|^{2}_{L^{2}(\Omega)}=B(\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh}),z_{kh})=B(P_{h}v_{k}-v_{kh},\Delta_{h}^{-2}z_{kh}).$

Using the orthogonality (18) of $v_{k}$ and $v_{kh}$ with respect to the
bilinear form $B$, we can replace $v_{kh}$ by $v_{k}$. Moreover, by definition
of $P_{h}$, all $L^{2}$ inner products vanish, leaving only the $H^{1}_{0}$
inner product:

$\mathcal{Z}=B(P_{h}v_{k}-v_{k},\Delta_{h}^{-2}z_{kh})=(\nabla(P_{h}v_{k}-v_{k}),\nabla(\Delta_{h}^{-2}z_{kh}))_{I\times\Omega}.$

In this inner product, we can replace $v_{k}$ by its Ritz projection
$R_{h}v_{k}$ and obtain after applying the definitions of $\Delta_{h}$,
$P_{h}$ and the duality pairing

$\displaystyle\mathcal{Z}=(\nabla(P_{h}v_{k}-R_{h}v_{k}),\nabla(\Delta_{h}^{-2}z_{kh}))_{I\times\Omega}$
$\displaystyle=-(P_{h}v_{k}-R_{h}v_{k},\Delta_{h}^{-1}z_{kh})_{I\times\Omega}=-(v_{k}-R_{h}v_{k},\Delta_{h}^{-1}z_{kh})_{I\times\Omega}$
$\displaystyle\leq\int_{I}\|(v_{k}-R_{h}v_{k})(t)\|_{H^{-1}(\Omega)}\|\Delta_{h}^{-1}z_{kh}(t)\|_{H^{1}_{0}(\Omega)}\
dt.$

The second term in the integral for each fixed $t$, can be estimated as
follows,

$\displaystyle\|\Delta_{h}^{-1}z_{kh}(t)\|_{H^{1}_{0}(\Omega)}^{2}$
$\displaystyle\leq
C\left(\nabla\Delta_{h}^{-1}z_{kh}(t),\nabla\Delta_{h}^{-1}z_{kh}(t)\right)_{\Omega}$
$\displaystyle=-C\left(z_{kh}(t),\Delta_{h}^{-1}z_{kh}(t)\right)_{\Omega}$
$\displaystyle\leq
C\|z_{kh}(t)\|_{L^{2}(\Omega)}\|\Delta_{h}^{-1}z_{kh}(t)\|_{L^{2}(\Omega)}$
$\displaystyle\leq
C\|z_{kh}(t)\|_{L^{2}(\Omega)}\|\Delta_{h}^{-1}z_{kh}(t)\|_{H^{1}_{0}(\Omega)},$

yielding $\|\Delta_{h}^{-1}z_{kh}(t)\|_{H^{1}_{0}(\Omega)}\leq
C\|z_{kh}(t)\|_{L^{2}(\Omega)}$ for almost all $t$. Using this estimate
together with Lemma 6.5, we get

$\mathcal{Z}\leq Ch^{3}\int_{I}\|\Delta
v_{k}(t)\|_{L^{2}(\Omega)}\|z_{kh}(t)\|_{L^{2}(\Omega)}\ dt\leq Ch^{3}\|\Delta
v_{k}\|_{L^{1}(I;L^{2}(\Omega))}\|z_{kh}\|_{L^{\infty}(I;L^{2}(\Omega))}.$

Using Corollary 4.2, we finally obtain

$\displaystyle\mathcal{Z}$ $\displaystyle\leq
Ch^{3}\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}\|z_{kh}(T)\|_{L^{2}(\Omega)}\leq
Ch^{3}\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}\|\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)}.$

Canceling $\|\Delta_{h}^{-2}(P_{h}v_{k}-v_{kh})(T)\|_{L^{2}(\Omega)}$ gives
the result.

Using this auxiliary result, we can prove the next lemmas estimating
$R_{h}v_{k}-v_{kh}$.

######  Lemma 6.11.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X_{k,h}^{r,2}$ be the semidiscrete and
fully discrete solutions of (9) and (17), respectively for $v_{0}\in
L^{2}(\Omega)$. There exists a constant $C$ independent of $k$,$h$, and $T$
such that

$\|\left(R_{h}v_{k}-v_{kh}\right)(T)\|_{{L^{2}(\Omega)}}\leq
Ch^{3}\left(\frac{1}{T^{2}}+\frac{1}{T^{\frac{3}{2}}}\right)\ln{\frac{T}{k}}\|v_{0}\|_{L^{2}(\Omega)}.$

######  Proof 6.12.

Let $y_{kh}\in X_{k,h}^{r,2}$ be the solution to a dual problem with
$y_{kh}(T)=\left(R_{h}v_{k}-v_{kh}\right)(T)$, i.e $y_{kh}\in X_{k,h}^{r,2}$
satisfies

$B(\varphi_{kh},y_{kh})=\left(\varphi_{kh}(T),(R_{h}v_{k}-v_{kh})(T)\right)\quad\text{for
all}\quad\varphi_{kh}\in X_{k,h}^{r,2}.$

To simplify notation, we define $\psi_{kh}:=R_{h}v_{k}-v_{kh}\in
X_{k,h}^{r,2}$. We introduce $\tilde{\psi}_{kh}\in X_{k,h}^{r,2}$ to be zero
on $I_{1}\cup...\cup I_{\tilde{m}}$ and $\tilde{\psi}_{kh}=\psi_{kh}$ on
$I_{\tilde{m}+1}\cup...I_{M}$ for $\tilde{m}$ chosen such that $\frac{T}{2}\in
I_{\tilde{m}}$. Analogously we define $\tilde{y}_{kh}$. Choosing
$\tilde{\psi}_{kh}$ as test function in the definition of $y_{kh}$ and
transfering the cutoff from one argument to the other, by (19), we get

$\displaystyle\|\left(R_{h}v_{k}-v_{kh}\right)(T)\|_{{L^{2}(\Omega)}}^{2}$
$\displaystyle=B(\tilde{\psi}_{kh},y_{kh})$
$\displaystyle=B(\psi_{kh},\tilde{y}_{kh})+\left(\psi_{kh,\tilde{m}}^{-},y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=B(R_{h}v_{k}-v_{kh},\tilde{y}_{kh})+\left((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=B(R_{h}v_{k}-v_{k},\tilde{y}_{kh})+\left((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=J_{1}+J_{2}.$

Here we also have used the Galerkin orthogonality (18) with respect to the
bilinear form $B$. By the definition of the Ritz projection the terms
$\left(\nabla(R_{h}v_{k}-v_{k}),\nabla\tilde{y}_{kh}\right)_{I_{m}\times\Omega}$
vanish from the form $B$, such that the remaining terms in $J_{1}$ are

$\displaystyle J_{1}$
$\displaystyle=-\smashoperator[]{\sum_{m=\tilde{m}+1}^{M}}(R_{h}v_{k}-v_{k},\partial_{t}y_{kh})_{I_{m}\times\Omega}-\smashoperator[]{\sum_{m=\tilde{m}+1}^{M}}(R_{h}v_{k,m}^{-}-v_{k,m}^{-},[y_{kh}]_{m})-\left(R_{h}v_{k,\tilde{m}}^{-}-v_{k,\tilde{m}}^{-},y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle\leq\|R_{h}v_{k}-v_{k}\|_{L^{\infty}((t_{\tilde{m}-1},T);{L^{2}(\Omega)})}\left(\|\partial_{t}y_{kh}\|_{L^{1}(I;{L^{2}(\Omega)})}+\smashoperator[]{\sum_{m=1}^{M}}\|[y_{kh}]_{m}\|_{{L^{2}(\Omega)}}+\|y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}\right),$

where we used the dual form of $B(\cdot,\cdot)$ and Hölders inequality in
space and time to estimate the terms. Applying Corollary 4.2, gives

$\|\partial_{t}y_{kh}\|_{L^{1}(I;{L^{2}(\Omega)})}+\sum_{m=1}^{M}\|[y_{kh}]_{m}\|_{{L^{2}(\Omega)}}+\|y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}\leq
C\ln{\frac{T}{k}}\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

Note that $y_{kh}$ is a solution to a dual problem and we use $y_{kh}(T)$ as
bound. Using the $L^{2}$ error estimate for the Ritz projection, together with
the estimate (47) of Lemma 6.3, we obtain for any $t\in(t_{\tilde{m}-1},T]$

$\ \|(R_{h}v_{k}-v_{k})(t)\|_{{L^{2}(\Omega)}}\leq
Ch^{3}\|v_{k}(t)\|_{H^{3}(\Omega)}\leq Ch^{3}\|\Delta
v_{k}(t)\|_{{L^{2}(\Omega)}}^{\frac{1}{2}}\|\Delta^{2}v_{k}(t)\|_{{L^{2}(\Omega)}}^{\frac{1}{2}}.$

Using the smoothing results of Lemma 4.6, we obtain

$\displaystyle\sup_{t\in(t_{\tilde{m}-1},T]}\|(R_{h}v_{k}-v_{k})(t)\|_{{L^{2}(\Omega)})}$
$\displaystyle\leq Ch^{3}\sup_{t\in(t_{\tilde{m}-1},T]}\|\Delta
v_{k}(t)\|_{{L^{2}(\Omega)}}^{\frac{1}{2}}\|\Delta^{2}v_{k}(t)\|_{{L^{2}(\Omega)}}^{\frac{1}{2}}$
(55) $\displaystyle\leq
C\dfrac{h^{3}}{t_{\tilde{m}}^{\frac{3}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\leq
C\dfrac{h^{3}}{T^{\frac{3}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}.$

In the last step, we used the estimate
$\frac{1}{t_{\tilde{m}}}\leq\frac{2}{T}$ which holds true, since
$t_{\tilde{m}}$ was chosen such that $\frac{T}{2}\in I_{\tilde{m}}$, and thus,
$\frac{T}{2}\leq t_{\tilde{m}}$. Combining these results gives the proposed
estimate for $J_{1}$:

$J_{1}\leq
C\dfrac{h^{3}}{T^{\frac{3}{2}}}\ln{\frac{T}{k}}\|v_{0}\|_{{L^{2}(\Omega)}}\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

To estimate $J_{2}$ we insert an artificial zero by adding and subtracting
$v_{k,\tilde{m}}^{-}$,

$\displaystyle J_{2}$
$\displaystyle=((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+})=((R_{h}v_{k,\tilde{m}}-v_{k,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+})+((v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+}):=J_{21}+J_{22}.$

The term $J_{21}$ can be estimated with (55), the discrete smoothing result of
Lemma 4.6 applied to $\|y_{kh,\tilde{m}}^{+}\|_{L^{2}(\Omega)}$ and the
special choice of $\tilde{m}$:

$\displaystyle J_{21}$
$\displaystyle\leq\sup_{t\in(t_{\tilde{m}-1},T)}\|(R_{h}v_{k}-v_{k})(t)\|_{{L^{2}(\Omega)})}\|y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$
$\displaystyle\leq
C\dfrac{h^{3}}{t_{\tilde{m}}^{\frac{3}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\|y_{kh}(T)\|_{{L^{2}(\Omega)}}\leq
C\dfrac{h^{3}}{T^{\frac{3}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

To estimate $J_{22}$ we use Lemma 6.9 by using $t_{\tilde{m}}$ as artificial
endtime. Here it is of importance, that the derived constant does not depend
on the endtime, since we need to replace $t_{\tilde{m}}$ by $T$ later. This
can only be done, when the explicit dependence of the result of Lemma 6.9 on
the endtime is known. We thus get after inserting the $L^{2}$ projection
operator:

$\displaystyle J_{22}$
$\displaystyle=((P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},y_{kh,\tilde{m}}^{+})$
$\displaystyle=(\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}^{2}y_{kh,\tilde{m}}^{+})$
$\displaystyle\leq\|\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-}\|_{{L^{2}(\Omega)}}\|\Delta_{h}^{2}y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$
$\displaystyle\leq
C\ln\left(\frac{t_{\tilde{m}}}{k}\right)\|v_{0}\|_{{L^{2}(\Omega)}}\frac{h^{3}}{{(T-t_{\tilde{m}})}^{2}}\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

In the last step, we have used Lemma 6.9 for
$\|\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-}\|_{{L^{2}(\Omega)}}$
and the discrete smoothing result of Lemma 4.6 for
$\|\Delta_{h}^{2}y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$. Since $y_{kh}$ is
a solution to a backwards problem, we use $\frac{1}{T-t_{\tilde{m}}}$, instead
of $\frac{1}{t_{\tilde{m}}}$. We now replace all occurrences of
$t_{\tilde{m}}$ by $T$. As before, we use $\frac{T}{2}\in I_{\tilde{m}}$
yielding $t_{\tilde{m}}\leq\frac{T}{2}+k$. For fine enough time
discretizations (i.e. $\frac{T}{4}>k$) we have

$T-t_{\tilde{m}}\geq T-\frac{T}{2}-k\geq
T-\frac{T}{2}-\frac{T}{4}=\frac{T}{4},$

thus giving $\frac{1}{T-t_{\tilde{m}}}\leq C\frac{1}{T}$. To estimate the
logarithmic term, we use the following consideration: Let $x\in\mathbb{R}$
such that $x>2$. Then it holds $x+1\leq x^{2}$. With the monotonicity of the
logarithm, we obtain $\ln(x+1)\leq\ln(x^{2})=2\ln(x)$. Applying this to the
logarithmic term, wile using $t_{\tilde{m}}\leq\frac{T}{2}+k$ and
$\frac{T}{2k}>2$, yields

$\ln\left(\frac{t_{\tilde{m}}}{k}\right)\leq\ln\left(\frac{\frac{T}{2}+k}{k}\right)=\ln\left(\frac{T}{2k}+1\right)\leq
2\ln\left(\frac{T}{2k}\right)\leq 2\ln\left(\frac{T}{k}\right).$

This gives a bound for $J_{22}$, depending on the final time $T$,

$J_{22}\leq
C\frac{h^{3}}{T^{2}}\ln{\frac{T}{k}}\|v_{0}\|_{{L^{2}(\Omega)}}\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

Dividing all considered terms by
$\|(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}$ gives the proposed estimate.

We now show a similar result for the discrete Laplacian.

######  Lemma 6.13.

Let $v_{k}\in X_{k}^{r}$ and $v_{kh}\in X_{k,h}^{r,2}$ be the semidiscrete and
fully discrete solutions of (9) and (17), respectively. Then there exists a
constant independent of $k$, $h$, and $T$ such that

$\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}\leq
Ch^{3}\left(\frac{1}{T^{3}}+\frac{1}{T^{\frac{5}{2}}}\right)\ln{\frac{T}{k}}\|v_{0}\|_{{L^{2}(\Omega)}}.$

######  Proof 6.14.

Let $y_{kh}\in X_{k,h}^{r,2}$ be the solution to a dual problem with
$y_{kh}(T)=\Delta_{h}\left(R_{h}v_{k}-v_{kh}\right)(T)$, i.e $y_{kh}\in
X_{k,h}^{r,2}$ satisfies

$B(\varphi_{kh},y_{kh})=\left(\varphi_{kh}(T),\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\right)\quad\text{for
all}\quad\varphi_{kh}\in X_{k,h}^{r,2}.$ (56)

As in the previous lemma, in order to simplify notation, we define
${\psi_{kh}:=R_{h}v_{k}-v_{kh}\in X_{k,h}^{r,2}}$. We introduce
$\tilde{\psi}_{kh}\in X_{k,h}^{r,2}$ to be zero on $I_{1}\cup...\cup
I_{\tilde{m}}$ and $\tilde{\psi}_{kh}=\psi_{kh}$ on
$I_{\tilde{m}+1}\cup...I_{M}$ for $\tilde{m}$ chosen such that $\frac{T}{2}\in
I_{\tilde{m}}$. We define $\tilde{y}_{kh}$ analogously. Choosing
$\Delta_{h}\tilde{\psi}_{kh}$ as test function in the definition of $y_{kh}$,
and transfering the cutoff from the first argument of $B$ to the second, by
applying (19), we get

$\displaystyle\|\Delta_{h}\left(R_{h}v_{k}-v_{kh}\right)(T)\|_{{L^{2}(\Omega)}}^{2}$
$\displaystyle=B(\Delta_{h}\tilde{\psi}_{kh},y_{kh})$
$\displaystyle=B(\tilde{\psi}_{kh},\Delta_{h}y_{kh})$
$\displaystyle=B(\psi_{kh},\Delta_{h}\tilde{y}_{kh})+\left(\psi_{kh,\tilde{m}}^{-},\Delta_{h}y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=B(R_{h}v_{k}-v_{kh},\Delta_{h}\tilde{y}_{kh})+\left((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=B(R_{h}v_{k}-v_{k},\Delta_{h}\tilde{y}_{kh})+\left((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+}\right)$
$\displaystyle=J_{1}+J_{2}.$

Here we have used the Galerkin orthogonality (18) with respect to the bilinear
form $B$. By the definition of the Ritz projection the terms
$\left(\nabla(R_{h}v_{k}-v_{k}),\nabla\Delta_{h}\tilde{y}_{kh}\right)_{I_{m}\times\Omega}$
vanish from the form $B$, such that the remaining terms of $J_{1}$ are

$J_{1}=\sum_{m=\tilde{m}+1}^{M}(\partial_{t}(R_{h}v_{k}-v_{k}),\Delta_{h}y_{kh})_{I_{m}\times\Omega}+\sum_{m=\tilde{m}+1}^{M}([R_{h}v_{k}-v_{k}]_{m},\Delta_{h}y_{kh,m}^{+}).$

Applying Hölder’s inequality in space and time gives

$\displaystyle
J_{1}\leq\|\partial_{t}(R_{h}v_{k}-v_{k})\|_{L^{\infty}((t_{\tilde{m}},T);{L^{2}(\Omega)})}\|\Delta_{h}y_{kh}\|_{L^{1}((t_{\tilde{m}},T);{L^{2}(\Omega)})}+\sum_{m=\tilde{m}+1}^{M}\|[R_{h}v_{k}-v_{k}]_{m}\|_{{L^{2}(\Omega)}}\|\Delta_{h}y_{kh,m}^{+}\|_{{L^{2}(\Omega)}}.$

Introducing an artificial factor $1$ as $k_{m}\cdot k_{m}^{-1}$ in the sum
allows us to extract the term $\max_{\tilde{m}\leq m\leq
M}\left\\{k_{m}^{-1}\|[R_{h}v_{k}-v_{k}]_{m}\|_{{L^{2}(\Omega)}}\right\\}$ out
of the sum. This gives

$\displaystyle J_{1}\leq$ $\displaystyle\max_{\tilde{m}\leq m\leq
M}\|\partial_{t}(R_{h}v_{k}-v_{k})\|_{L^{\infty}(I_{m};{L^{2}(\Omega)})}\|\Delta_{h}y_{kh}\|_{L^{1}((t_{\tilde{m}},T);{L^{2}(\Omega)})}$
$\displaystyle+\max_{\tilde{m}\leq m\leq
M}\left\\{k_{m}^{-1}\|[R_{h}v_{k}-v_{k}]_{m}\|_{{L^{2}(\Omega)}}\right\\}\left(\sum_{m=\tilde{m}+1}^{M}k_{m}\|\Delta_{h}y_{kh,m}^{+}\|_{{L^{2}(\Omega)}}\right).$

Using Corollary 4.2 and the $L^{2}$ error estimate for the Ritz projection for
the other terms, gives the following estimate,

$\displaystyle J_{1}\leq Ch^{3}\left(\max_{\tilde{m}\leq m\leq
M}\|\partial_{t}v_{k}\|_{L^{\infty}(I_{m};H^{3}(\Omega))}+\max_{\tilde{m}\leq
m\leq
M}\left\\{k_{m}^{-1}\|[v_{k}]_{m}\|_{H^{3}(\Omega))}\right\\}\right)\ln{\frac{T}{k}}\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

Similar to the previous lemma, by the estimate (47) of Lemma 6.3 and Remark
6.1, yielding $\partial_{t}\Delta v_{k}(t)\in H^{1}_{0}(\Omega)$ for
$t\in(t_{\tilde{m}-1},T]$, we obtain

$\|\partial_{t}v_{k}(t)\|_{H^{3}(\Omega)}\leq C\|\partial_{t}\Delta
v_{k}(t)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}\|\partial_{t}\Delta^{2}v_{k}(t)\|^{\frac{1}{2}}_{{L^{2}(\Omega)}}.$

Taking the supremum over $(t_{\tilde{m}-1},T]$ and using Lemma 4.6 then yields

$\displaystyle\max_{\tilde{m}\leq m\leq
M}\|\partial_{t}v_{k}(t)\|_{L^{\infty}(I_{m};H^{3}(\Omega))}$
$\displaystyle\leq C\max_{\tilde{m}\leq m\leq M}\left\|\partial_{t}\Delta
v_{k}(t)\right\|^{\frac{1}{2}}_{L^{\infty}(I_{m};{L^{2}(\Omega)})}\left\|\partial_{t}\Delta^{2}v_{k}(t)\right\|^{\frac{1}{2}}_{L^{\infty}(I_{m};{L^{2}(\Omega)})}$
$\displaystyle\leq
C\dfrac{1}{t_{\tilde{m}}^{\frac{5}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\leq
C\dfrac{1}{T^{\frac{5}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}.$

Applying the same arguments, using $\Delta v_{k}(t)\in H^{1}_{0}(\Omega)$ for
$t\in(t_{\tilde{m}-1},T)$, and Lemma 4.6 gives

$\displaystyle\max_{\tilde{m}\leq m\leq
M}\left\\{k_{m}^{-1}\|[v_{k}]_{m}\|_{H^{3}(\Omega)}\right\\}\leq\max_{\tilde{m}\leq
m\leq
M}\left\\{k_{m}^{-1}\left(\|\Delta[v_{k}]_{m}\|_{{L^{2}(\Omega)}}\|\Delta^{2}[v_{k}]_{m}\|_{{L^{2}(\Omega)}}\right)^{\frac{1}{2}}\right\\}\leq
C\dfrac{1}{t_{\tilde{m}}^{\frac{5}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\leq
C\dfrac{1}{T^{\frac{5}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}.$

Summarizing all above results yields the final bound for $J_{1}$:

$J_{1}\leq
Ch^{3}\dfrac{1}{T^{\frac{5}{2}}}\ln{\frac{T}{k}}\|v_{0}\|_{{L^{2}(\Omega)}}\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

To estimate $J_{2}$ we insert an artificial zero like before by adding and
subtracting $v_{k,\tilde{m}}^{-}$:

$\displaystyle J_{2}$
$\displaystyle=((R_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+})=((R_{h}v_{k,\tilde{m}}-v_{k,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+})+((v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+}):=J_{21}+J_{22}.$

The term $J_{21}$ can be estimated similarly to the previous lemma, applying
(55), the discrete smoothing result of Lemma 4.6 for
$\|\Delta_{h}y_{kh,\tilde{m}}^{+}\|_{L^{2}(\Omega)}$ and using the special
choice of $\tilde{m}$:

$\displaystyle J_{21}$
$\displaystyle\leq\sup_{t\in(t_{\tilde{m}-1},T)}\|(R_{h}v_{k}-v_{k})(t)\|_{{L^{2}(\Omega)}}\|\Delta_{h}y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$
$\displaystyle\leq
C\dfrac{h^{3}}{t_{\tilde{m}}^{\frac{3}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\frac{1}{T-t_{\tilde{m}}}\|y_{kh}(T)\|_{{L^{2}(\Omega)}}$
$\displaystyle\leq
C\dfrac{h^{3}}{T^{\frac{5}{2}}}\|v_{0}\|_{{L^{2}(\Omega)}}\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

We estimate $J_{22}$ by replacing $v_{k,\tilde{m}}$ with its
$L^{2}$-projection:

$\displaystyle J_{22}$
$\displaystyle=((P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}y_{kh,\tilde{m}}^{+})$
$\displaystyle=(\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-},\Delta_{h}^{3}y_{kh,\tilde{m}}^{+})$
$\displaystyle\leq\|\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-}\|_{{L^{2}(\Omega)}}\|\Delta_{h}^{3}y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$
$\displaystyle\leq
C\dfrac{h^{3}}{(T-t_{\tilde{m}})^{3}}\ln\left(\frac{t_{\tilde{m}}}{k}\right)\|v_{0}\|_{{L^{2}(\Omega)}}\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

In the last step, we have used Lemma 6.9 for
$\|\Delta_{h}^{-2}(P_{h}v_{k,\tilde{m}}-v_{kh,\tilde{m}})^{-}\|_{{L^{2}(\Omega)}}$
and the discrete smoothing result of Lemma 4.6 for
$\|\Delta_{h}^{3}y_{kh,\tilde{m}}^{+}\|_{{L^{2}(\Omega)}}$. Since $y_{kh}$ is
the solution to a problem backward in time, we use $\frac{1}{T-t_{\tilde{m}}}$
instead of $\frac{1}{t_{\tilde{m}}}$ in the application of this result.
Analogously to the previous lemma, we can replace the terms involving
$t_{\tilde{m}}$ by ones dependent only of $T$ because of the special choice of
$t_{\tilde{m}}$, thus giving the final bound for $J_{22}$,

$J_{22}\leq
C\frac{h^{3}}{T^{3}}\ln{\frac{T}{k}}\|v_{0}\|_{{L^{2}(\Omega)}}\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}.$

Dividing all considered terms by
$\|\Delta_{h}(R_{h}v_{k}-v_{kh})(T)\|_{{L^{2}(\Omega)}}$ gives the proposed
estimate.

Combining Lemma 6.11 and Lemma 6.13 with the discrete Gagliardo-Nirenberg
inequality (30) gives the following result: Let $v_{k}\in X_{k}^{r}$ and
$v_{kh}\in X_{k,h}^{r,2}$ be the semidiscrete and fully discrete solutions of
of (1) with $v_{0}\in L^{2}(\Omega)$. Then there exists a constant independent
of $k$ and $h$ such that

$\|(R_{h}v_{k}-v_{kh})(T)\|_{L^{\infty}(\Omega)}\leq
Ch^{3}\ln{\frac{T}{k}}\left(\frac{1}{T^{3}}+\frac{1}{T^{\frac{5}{2}}}\right)^{\frac{N}{4}}\left(\frac{1}{T^{2}}+\frac{1}{T^{\frac{3}{2}}}\right)^{\left(1-\frac{N}{4}\right)}\|v_{0}\|_{{L^{2}(\Omega)}}.$

This result now allows us to estimate the final term $T_{3}$ of (50) and thus
proves Theorem 6.7.

###  6.2. Estimates for $(v-v_{kh})(T)$ with $v_{0}\in\mathcal{M}(\Omega)$

Now that we have established Theorem 6.7 for $v_{0}\in L^{2}(\Omega)$,
following exactly the proof of Theorem 5.2, and using the Assumption 1, we can
establish

######  Theorem 6.15.

Let $\Omega_{0}$ be a subdomain with $\overline{\Omega}_{0}\subset\Omega$,
$v_{0}\in\mathcal{M}(\Omega)$ with
$\operatorname{supp}v_{0}\subset\Omega_{0}$. Let $v$ and $v_{kh}\in
X_{k,h}^{r,2}$ satisfy (1) and (17), respectively. In addition, let $\Omega$
be such that Assumption 1 holds.

$\|(v-v_{kh})(T)\|_{L^{\infty}(\Omega_{0})}\leq
C(T,\Omega_{0})\left(\ell_{k}h^{3}+k^{2r+1}\right)\|v_{0}\|_{\mathcal{M}(\Omega)},$
(57)

where $\ell_{k}=\ln{\frac{T}{k}}$,
$d=\operatorname*{dist}(\Omega_{0},\partial\Omega)$ and $C(T,d)$ is a constant
depending on $T$ and $d$.

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