A discontinuous Galerkin method for elliptic-hyperbolic equations

TL;DR

This paper develops and analyzes a discontinuous Galerkin method tailored for second-order mixed-type PDEs that transition from elliptic to hyperbolic, providing theoretical error estimates and numerical validation.

Contribution

It introduces a novel DG method for mixed-type equations, establishing well-posedness, deriving $hp$-a priori error estimates, and validating results through numerical experiments.

Findings

The method is well-posed via coercivity in an energy norm.

Derived $hp$-a priori error estimates for convergence.

Numerical experiments confirm theoretical convergence rates.

Abstract

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.