The Optimal L2-Norm Error Estimate of a Weak Galerkin Finite Element Method for a Multi-Dimensional Evolution Equation with a Weakly Singular Kernel

Abstract

This paper proposes a weak Galerkin (WG) finite element method for solving a multi-dimensional evolution equation with a weakly singular kernel. The temporal discretization employs the backward Euler scheme, while the fractional integral term is approximated via a piecewise constant function method. A fully discrete scheme is constructed by integrating the WG finite element approach for spatial discretization. $L^2$-norm stability and convergence analysis of the fully discrete scheme are rigorously established. Numerical experiments are conducted to validate the theoretical findings and demonstrate optimal convergence order in both spatial and temporal directions. The numerical results confirm that the proposed method achieves an accuracy of the order $O\left(\tau +h^{k+1}\right)$, where $\tau$ and h represent the time step and spatial mesh size, respectively. This work extends previous studies on one-dimensional problems to higher spatial dimensions, providing a robust framework for handling evolution equations with a weakly singular kernel.