Convection-adapted BEM-based FEM

TL;DR

This paper introduces a convection-adapted BEM-based FEM for 3D convection-diffusion-reaction problems, utilizing PDE-harmonic shape functions and local boundary element techniques to enhance stability in convection-dominated scenarios.

Contribution

It develops a novel non-standard finite element method with PDE-harmonic shape functions and a convection-adapted boundary data procedure, improving stability over traditional methods.

Findings

Enhanced stability in convection-dominated problems

Effective use of local boundary element techniques

Numerical experiments demonstrate improved performance

Abstract

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.