Numerical Analysis of Stabilizer‐Free Weak Galerkin Finite Element Method for Time‐Dependent Differential Equation Under Low Regularity

ABSTRACT In this article, we focus on investigating the optimal convergence order of error estimates for the stabilizer‐free weak Galerkin finite element method for a second‐order time‐dependent differential equation under low regularity assumptions. In most of the existing literature on stabilizer‐free weak Galerkin finite element methods, the exact solution is usually assumed to have at least regularity so as to achieve convergence rates. But in many applications, the exact solution has only regularity, and hence the existing error analysis is no longer applicable. Our objective is to build a theoretical framework for filling in this gap. Our key idea is to interpolate an approximation between the exact solution and the stabilizer‐free weak Galerkin approximation as an intermediate auxiliary step. Based on this, the optimal convergence order of error estimates is proved to be of . Finally, we present several numerical experiments on uniform triangulations and square partitions to demonstrate the theoretical convergence properties.