Weak discrete maximum principle and $L^\infty$ analysis of the DG method for the Poisson equation on a polygonal domain
Provides rigorous L∞ error analysis for DG methods on polygonal domains, which is a theoretical contribution for numerical analysts working on finite element methods.
The authors derive L∞ error estimates for the symmetric interior penalty discontinuous Galerkin method for the Poisson equation on polygonal domains, establishing a weak maximum principle for discrete harmonic functions. Numerical examples validate the theoretical results.
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.