Error estimates in weak Galerkin finite element methods for parabolic equations under low regularity assumptions

Abstract

In this paper, we consider the weak Galerkin finite element approximations of second order linear parabolic problems in two dimensional convex polygonal domains under the low regularities of the solutions. Optimal order error estimates in L2(L2) and L2(H1) norms are shown to hold for both the spatially discrete continuous time and the discrete time weak Galerkin finite element schemes, which allow using the discontinuous piecewise polynomials on finite element partitions with the arbitrary shape of polygons with certain shape regularity. The fully discrete scheme is based on first order in time Euler method. We have derived O(hr+1) in L2(L2) norm and O(hr) in L2(H1) norm when the exact solution u∈L2(0,T;Hr+1(Ω))∩H1(0,T;Hr−1(Ω)), for some r≥1. Numerical experiments are reported for several test cases to justify our theoretical convergence results.