Stabilizer-Free Weak Galerkin Methods for Monotone Quasilinear Elliptic PDEs

TL;DR

This paper introduces stabilizer-free weak Galerkin methods for solving monotone quasilinear elliptic PDEs, providing theoretical guarantees of existence, uniqueness, and optimal error estimates, validated by numerical experiments.

Contribution

The paper develops a novel stabilizer-free weak Galerkin approach for nonlinear elliptic PDEs, ensuring stability and optimal convergence without additional stabilization terms.

Findings

Unique solvability of the discrete problem

Optimal error estimates in the energy norm

Numerical results confirming theoretical predictions

Abstract

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.