Fully Discrete Pointwise Smoothing Error Estimates for Measure Valued Initial Data

Problem Statement
This paper addresses the numerical analysis of a homogeneous parabolic problem defined on a bounded, convex, polygonal/polyhedral domain $\Omega \subset \mathbb{R}^N$ ($N=2,3$) over a time interval $I=(0, T]$. The specific focus is on the case where the initial data $v_0$ is a regular Borel measure ($v_0 \in \mathcal{M}(\Omega)$), which may include singularities such as Dirac delta functions. The authors investigate the fully discrete approximation of this problem using a discontinuous Galerkin (dG) method of arbitrary degree $r \ge 0$ in time and continuous finite elements of order $s=1$ (linear) or $s=2$ (quadratic) in space.

The primary objective is to derive pointwise ($L^\infty$) error estimates for the numerical solution evaluated at the final time $T$. Specifically, the authors aim to establish error bounds of the form:
$$\|(v - v_{kh})(T)\|_{L^\infty(\Omega_0)} \le C(\Omega_0, T) \left( k^{2r+1} + \ell_{kh} h^{s+1} \right) \|v_0\|_{\mathcal{M}(\Omega)}$$
where $\Omega_0$ is a subdomain containing the support of the initial measure, $k$ is the time step, $h$ is the spatial mesh size, and $\ell_{kh}$ is a logarithmic term.

Methodology
The analysis proceeds through several stages of discretization and regularity theory:

1. Very Weak Solutions: The authors first establish the existence and regularity of "very weak" solutions for the continuous problem with measure-valued initial data. They utilize a duality argument and specific test function spaces to handle the lack of regularity in $v_0$.
2. Discretization Schemes:
   • Time: A discontinuous Galerkin (dG) scheme is employed. For measure-valued data, the trial and test spaces are carefully modified (using $W^{1,p}_0$ spaces) to ensure the initial data pairing is well-defined, as standard $H^1_0$ spaces do not embed into continuous functions for $N \ge 2$.
   • Space: Continuous Lagrange finite elements of degree $s$ are used.
3. Smoothing Estimates: A critical component of the methodology is the derivation of parabolic smoothing estimates. The authors prove that both the continuous and discrete (semidiscrete and fully discrete) solutions exhibit smoothing properties, mapping the rough initial measure data into $L^\infty$ spaces as time evolves. These estimates are derived for both $L^2$ and measure-valued initial data, showing decay rates dependent on time $t$ and the dimension $N$.
4. Error Decomposition: The total error is split into three components:
   • Time discretization error ($v - v_k$).
   • Spatial discretization error of the semidiscrete solution ($v_k - R_h v_k$).
   • Fully discrete error ($R_h v_k - v_{kh}$).
     The authors utilize interior error estimates and duality arguments to bound these terms, particularly focusing on the case where the initial data is supported in a subdomain $\Omega_0 \subset \Omega$.

Key Contributions and Results

• Fully Discrete Pointwise Estimates for Measure Data: The main result (Theorem 5.3) establishes fully discrete $L^\infty$ error estimates for initial data in $\mathcal{M}(\Omega)$ using piecewise linear finite elements ($s=1$). The estimate holds without additional smoothness assumptions beyond the convexity of the domain. The error bound is $O(k^{2r+1} + \ell_{kh} h^2)$.
• Extension to Quadratic Elements: The paper extends these results to quadratic finite elements ($s=2$) in Section 6. This requires additional regularity assumptions on the domain (e.g., rectangular domains) to ensure $H^3$ regularity of the solution. Under these assumptions, the spatial convergence rate improves to $O(h^3)$, yielding an error bound of $O(k^{2r+1} + \ell_{kh} h^3)$.
• Interior Estimates: The results are "interior" in nature, meaning the $L^\infty$ error is estimated on a subdomain $\Omega_0$ where the initial measure is supported. This allows the authors to avoid global singularities that might arise at the boundary or outside the support of the measure.
• General Time Partitions: The analysis accommodates potentially highly varying time partitions, provided they satisfy specific quasi-uniformity and grading conditions (e.g., $k_{min} \ge c k^\beta$).

Significance and Motivation
The authors state that these results are motivated by the need for sharp error estimates in initial data identification problems for parabolic equations, specifically when observations are taken at a finite number of points at the final time. In such inverse problems, the adjoint state satisfies a backward-in-time problem with a final condition given by a measure supported on the observation points.

While previous work by the authors [15] established similar results for $L^2$ initial data and linear elements, the extension to measure-valued data is necessary to handle the specific structure of the adjoint problem in sparse control and pointwise observation scenarios. The paper claims to provide the first fully discrete pointwise error estimates for Galerkin methods applied to parabolic problems with measure-valued initial data on quasi-uniform meshes, covering both linear and quadratic spatial discretizations. The results are presented as essential for obtaining sharp convergence rates in optimal control and inverse problems involving sparse controls or pointwise measurements.