Discontinuous Galerkin methods for semilinear elliptic boundary value problem

TL;DR

This paper develops and analyzes a discontinuous Galerkin method for semilinear elliptic boundary value problems, providing theoretical error estimates and numerical validation of the approach's effectiveness.

Contribution

The paper introduces a novel DG scheme for semilinear elliptic problems with rigorous analysis and optimal error estimates, supported by numerical experiments.

Findings

Proved existence and uniqueness of the DG solution.

Established optimal a priori error estimates.

Numerical results confirm the method's efficiency.

Abstract

A discontinuous Galerkin (DG) scheme for solving semilinear elliptic problem is developed and analyzed in this paper. The DG finite element discretizations are established, and the corresponding existence and uniqueness theorem is proved by using Brouwer's fixed point method. Some optimal priori error estimates under both DG norm and $L^2$ norm are presented. Numerical results are also shown to confirm the efficiency of the proposed approach.