An $L^p$-weak Galerkin method for second order elliptic equations in non-divergence form

TL;DR

This paper introduces a novel primal-dual weak Galerkin method for second order elliptic equations in non-divergence form, using an $L^p$-optimization framework with proven convergence and optimal error estimates.

Contribution

It develops a new $L^p$-weak Galerkin method with a constrained optimization approach, including an iterative algorithm and convergence analysis, for non-divergence form elliptic equations.

Findings

Proved existence and uniqueness of the numerical solution.

Established optimal error estimates in multiple norms.

Validated the method through numerical experiments with smooth and non-smooth coefficients.

Abstract

This article presents a new primal-dual weak Galerkin method for second order elliptic equations in non-divergence form. The new method is devised as a constrained $L^p$-optimization problem with constraints that mimic the second order elliptic equation by using the discrete weak Hessian locally on each element. An equivalent min-max characterization is derived to show the existence and uniqueness of the numerical solution. Optimal order error estimates are established for the numerical solution under the discrete $W^{2,p}$ norm, as well as the standard $W^{1,p}$ and $L^p$ norms. An equivalent characterization of the optimization problem in term of a system of fixed-point equations via the proximity operator is presented. An iterative algorithm is designed based on the fixed-point equations to solve the optimization problems. Implementation of the iterative algorithm is studied and convergence of the iterative algorithm is established. Numerical experiments for both smooth and non-smooth coefficients problems are presented to verify the theoretical findings.