An efficient high-order weak Galerkin finite element approach for Sobolev equation with variable matrix coefficients

Abstract

This paper constructs a high-order weak Galerkin finite element method for solving a two-dimensional Sobolev equation with variable matrix coefficients subjects to initial and boundary conditions. The proposed approach approximates the exact solution in time using interpolation techniques whereas the space derivatives are approximated by weak forms through integration by parts. The new algorithm is unconditionally stable, temporal second order convergence and spatial accurate with convergence order O(hp+1), in the L∞(0,Tf;H1)-norm, where p is a nonnegative integer and h represents the grid space. The developed computational scheme is faster and more efficient than a broad range of numerical methods deeply studied in the literature for solving Sobolev problems. Some numerical examples are carried out to confirm the theory and to investigate the performance and validity of the constructed high-order numerical scheme.