A Priori Error Bounds for Parabolic Interface Problems with Measure Data

TL;DR

This paper derives a priori error bounds for finite element solutions of parabolic interface problems with low-regularity measure data, addressing challenges posed by the data's irregularity.

Contribution

It provides the first a priori error estimates in the $L^2(L^2)$-norm for such problems with minimal regularity assumptions and smooth interfaces.

Findings

Error bounds established in $L^2(L^2)$-norm

Applicable to problems with measure data and low regularity

Uses duality argument and $L^2$ projections

Abstract

This article studies a priori error analysis for linear parabolic interface problems with measure data in time in a bounded convex polygonal domain in $\mathbb{R}^2$. We have used the standard continuous fitted finite element discretization for the space. Due to the low regularity of the data of the problem, the solution possesses very low regularity in the entire domain. A priori error bound in the $L^2(L^2(\Omega))$-norm for the spatially discrete finite element approximations are derived under minimal regularity with the help of the $L^2$ projection operators and the duality argument. The interfaces are assumed to be smooth for our purpose.