A Nystr\"om-based finite element method on polygonal elements

TL;DR

This paper introduces a Nyström-based finite element method for polygonal meshes, enabling accurate approximation on complex geometries with non-polynomial data through integral equation discretization.

Contribution

It presents a novel finite element approach using Nyström discretizations for polygonal elements, accommodating curvilinear shapes and non-polynomial boundary conditions.

Findings

Demonstrates high approximation quality of the proposed method.

Validates the approach through several numerical experiments.

Supports complex polygonal geometries with non-polynomial data.

Abstract

We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nystr\"om discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces.