A Modified Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

TL;DR

This paper introduces a modified primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form, simplifying the system and improving computational efficiency while maintaining optimal error estimates.

Contribution

The paper proposes a simplified primal-dual weak Galerkin scheme that reduces degrees of freedom and enhances numerical stability for non-divergence form elliptic equations.

Findings

Reduced degrees of freedom in the simplified system.

Significantly lower condition number with appropriate bilinear term.

Optimal error estimates in multiple norms.

Abstract

A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. In addition, the condition number of the simplified system could be greatly reduced when a newly introduced bilinear term in the M-PDWG scheme is appropriately chosen. Optimal order error estimates are derived for the numerical approximations in the discrete $H^2$-norm, $H^1$-norm and $L^2$-norm respectively. Extensive numerical results are demonstrated for both the smooth and non-smooth coefficients on convex and non-convex domains to verify the accuracy of the theory developed in this paper.