Embedded Trefftz DG framework for the analysis of discretizations with local-global decompositions

TL;DR

This paper introduces a framework for analyzing discretization methods that decompose problems into local and global parts, providing error bounds for embedded Trefftz DG methods and quasi-Trefftz methods across various PDEs.

Contribution

It offers the first comprehensive error analysis for embedded Trefftz DG methods and extends the framework to quasi-Trefftz methods with optimal error bounds.

Findings

Error bounds for embedded Trefftz DG methods across multiple PDEs

Optimal error estimates for quasi-Trefftz methods in weaker norms

A unified framework for local-global decomposition analysis

Abstract

This paper presents a framework for the analysis of discretization methods based on the decomposition into local and global problems. We apply the framework to provide a comprehensive error analysis for the embedded Trefftz discontinuous Galerkin method, for a wide range of second-order scalar elliptic partial differential equations and a scalar reaction-advection problem. We also analyze quasi-Trefftz methods with our framework, presenting the first optimal error bounds in weaker norms.