A staggered discontinuous Galerkin method for a class of nonlinear elliptic equations

TL;DR

This paper introduces a staggered discontinuous Galerkin method tailored for nonlinear elliptic equations, demonstrating stability, superconvergence, and confirming theoretical error estimates through numerical experiments.

Contribution

The paper develops a novel SDG method for nonlinear elliptic equations, including stability analysis, error estimates, and superconvergence properties, with numerical validation.

Findings

Method achieves superconvergence after local post-processing.

Numerical results confirm theoretical convergence rates.

Stability and error estimates are rigorously derived.

Abstract

In this paper, we present a staggered discontinuous Galerkin (SDG) method for a class of nonlinear elliptic equations in two dimensions. The SDG methods have some distinctive advantages, and have been successfully applied to a wide range of problems including Maxwell equations, acoustic wave equation, elastodynamics and incompressible Navier-Stokes equations. Among many advantages of the SDG methods, one can apply a local post-processing technique to the solution, and obtain superconvergence. We will analyze the stability of the method and derive a priori error estimates. We solve the resulting nonlinear system using the Newton's method, and the numerical results confirm the theoretical rates of convergence and superconvergence.