Weak Galerkin Finite Element Methods for Parabolic Problems with $L^2$ Initial Data

Abstract

We analyze the weak Galerkin finite element methods for second-order linear parabolic problems with $L^2$ initial data, both in a spatially semidiscrete case and in a fully discrete case based on the backward Euler method. We have established optimal $L^2$ error estimates of order $O(h^2/t)$ for semidiscrete scheme. Subsequently, the results are extended for fully discrete scheme. The error analysis has been carried out on polygonal meshes for discontinuous piecewise polynomials in finite element partitions. Finally, numerical experiments confirm our theoretical convergence results and efficiency of the scheme.