A weak Galerkin least squares finite element method for linear convection equations in non-divergence form

TL;DR

This paper introduces a weak Galerkin least-squares finite element method for linear convection equations in non-divergence form, achieving optimal error estimates without coercivity assumptions, applicable to complex meshes.

Contribution

It develops a novel WG--LS method that handles non-divergence form equations with minimal regularity and produces symmetric positive definite systems.

Findings

Optimal-order error estimates are established.

Numerical experiments confirm theoretical convergence.

Method demonstrates high accuracy and efficiency.

Abstract

This article develops a weak Galerkin least-squares (WG--LS) finite element method for first-order linear convection equations in non-divergence form. The method is formulated using discontinuous finite element functions and does not require any coercivity assumption on the convection vector or reaction coefficient. The resulting discrete problem leads to a symmetric and positive definite linear system and is applicable to general polygonal and polyhedral meshes. Under minimal regularity assumptions on the coefficients, optimal-order error estimates are established for the WG--LS approximation in a suitable energy norm. Numerical experiments are presented to confirm the theoretical convergence results and to demonstrate the accuracy and efficiency of the proposed method.