Geometric Transformation of Finite Element Methods: Theory and Applications

TL;DR

This paper introduces a novel geometric transformation technique for finite element methods that simplifies solving PDEs on curved domains by transforming them into problems on polyhedral domains, enabling higher-order convergence.

Contribution

The paper develops a new approach using low-regularity coordinate transformations and the broken Bramble-Hilbert lemma to achieve higher-order finite element convergence on curved domains.

Findings

Transformation simplifies geometric complexity of curved domains

Higher-order convergence rates are theoretically proven

Numerical experiments confirm theoretical predictions

Abstract

We present a new technique to apply finite element methods to partial differential equations over curved domains. A change of variables along a coordinate transformation satisfying only low regularity assumptions can translate a Poisson problem over a curved physical domain to a Poisson problem over a polyhedral parametric domain. This greatly simplifies both the geometric setting and the practical implementation, at the cost of having globally rough non-trivial coefficients and data in the parametric Poisson problem. Our main result is that a recently developed broken Bramble-Hilbert lemma is key in harnessing regularity in the physical problem to prove higher-order finite element convergence rates for the parametric problem. Numerical experiments are given which confirm the predictions of our theory.