# Trefftz Finite Elements on Curvilinear Polygons

## TL;DR

This paper introduces a Trefftz finite element method for meshes with curvilinear polygons, utilizing integral equations for basis functions, and demonstrates optimal convergence even with singularities and curved edges.

## Contribution

It develops a novel Trefftz finite element approach on curved polygonal meshes, including new definitions of polynomial edge functions and an interpolation operator with proven optimal convergence.

## Key findings

- Finite element solutions achieve optimal order convergence.
- Method works effectively on meshes with curved edges.
- Exploits singular functions for improved convergence without adaptive refinement.

## Abstract

We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local finite element spaces, and discuss how to work with these edge polynomial spaces in practice. An interpolation operator is introduced for the resulting finite elements, and we prove that it provides optimal order convergence for interpolation error under reasonable assumptions. We provide a description of the integral equation approach used for the examples in this paper, which was recently developed precisely with these applications in mind. A few numerical examples illustrate this optimal order convergence of the finite element solution on some families of meshes in which every element has at least one curved edge. We also demonstrate that it is possible to exploit the approximation power of locally singular functions that may exist in our finite element spaces in order to achieve optimal order convergence without the typical adaptive refinement toward singular points.