# Weak discrete maximum principle and $L^\infty$ analysis of the DG method for the Poisson equation on a polygonal domain

## TL;DR

This paper establishes $L^
Infty$ error bounds and a weak maximum principle for the symmetric interior penalty DG method solving the Poisson equation on polygonal domains, supported by theoretical proofs and numerical validation.

## Contribution

It introduces new $L^
Infty$ error estimates and a weak maximum principle for the DG method applied to polygonal domains, enhancing understanding of stability and accuracy.

## Key findings

- $L^
Infty$ error estimates are derived for the DG method.
- A weak maximum principle for discrete harmonic functions is proved.
- Numerical examples confirm the theoretical results.

## Abstract

We derive several $L^\infty$ error estimates for the symmetric interior penalty (SIP) discontinuous Galerkin (DG) method applied to the Poisson equation in a two-dimensional polygonal domain. Both local and global estimates are examined. The weak maximum principle (WMP) for the discrete harmonic function is also established. We prove our $L^\infty$ estimates using this WMP and several $W^{2,p}$ and $W^{1,1}$ estimates for the Poisson equation. Numerical examples to validate our results are also presented.