A Least-Squares Weak Galerkin Method for Second-Order Elliptic Equations in Non-Divergence Form

TL;DR

This paper introduces a new least-squares weak Galerkin method for second-order elliptic equations in non-divergence form, providing a symmetric positive definite system and optimal error estimates validated by numerical experiments.

Contribution

The paper develops a novel LS-WG method that constructs a discrete weak Hessian operator, applicable to general meshes, with proven optimal error bounds and validated through numerical tests.

Findings

The method produces a symmetric positive definite linear system.

Optimal-order error estimates are established.

Numerical experiments confirm theoretical results and robustness.

Abstract

This article proposes a novel least-squares weak Galerkin (LS-WG) method for second-order elliptic equations in non-divergence form. The approach leverages a locally defined discrete weak Hessian operator constructed within the weak Galerkin framework. A key feature of the resulting algorithm is that it yields a symmetric and positive definite linear system while remaining applicable to general polygonal and polyhedral meshes. We establish optimal-order error estimates for the approximation in a discrete $H^2$-equivalent norm. Finally, comprehensive numerical experiments are presented to validate the theoretical analysis and demonstrate the efficiency and robustness of the method.