Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations

TL;DR

This paper introduces an adaptive discontinuous Galerkin method tailored for non-linear diffusion-convection-reaction equations, effectively handling sharp layers and oscillations with a new preconditioning approach.

Contribution

The work presents a novel adaptive DG method with an efficient preconditioner for solving complex non-linear PDEs with sharp features.

Findings

DGAFEM effectively dampens oscillations

Resolves sharp layers accurately

Preconditioner improves computational efficiency

Abstract

In this work, we apply the adaptive discontinuous Galerkin (DGAFEM) method to the convection dominated non-linear, quasi-stationary diffusion-convection-reaction equations. We propose an efficient preconditioner using a matrix reordering scheme to solve the sparse linear systems iteratively arising from the discretized non-linear equations. Numerical examples demonstrate effectiveness of the DGAFEM to damp the spurious oscillations and resolve well the sharp layers occurring in convection dominated non-linear equations.