Geometric Transformation of Finite Element Methods: Theory and Applications
This work simplifies finite element implementation for curved domains, benefiting computational scientists and engineers solving PDEs on complex geometries.
The paper introduces a geometric transformation technique for finite element methods that converts Poisson problems on curved domains into problems on polyhedral parametric domains, enabling higher-order convergence rates despite rough coefficients. Numerical experiments validate the theoretical predictions.
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.