A priori error bounds for parabolic interface problems with measure data

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