Numerical solution of the div-curl problem by finite element exterior calculus

TL;DR

This paper presents a finite element exterior calculus approach for numerically reconstructing vector fields with specified divergence and curl in complex domains, demonstrating its effectiveness through implementation and numerical experiments.

Contribution

It introduces a novel finite element exterior calculus method for div-curl problems in non-contractible domains, extending previous approaches and providing practical implementation insights.

Findings

Effective reconstruction of vector fields in complex domains

Implementation using FEniCS library demonstrates practicality

Numerical results validate the method's accuracy

Abstract

We are interested in the numerical reconstruction of a vector field with prescribed divergence and curl in a general domain of R 3 or R 2 , not necessarily contractible. To this aim, we introduce some basic concepts of finite element exterieur calculus and rely heavily on recent results of P. Leopardi and A. Stern. The goal of the paper is to take advantage of the links between usual vector calculus and exterior calculus and show the interest of the exterior calculus framework, without too much prior knowledge of the subject. We start by describing the method used for contractible domains and its implementation using the FEniCS library (see fenicsproject.org). We then address the problems encountered with non contractible domains and general boundary conditions and explain how to adapt the method to handle these cases. Finally we give some numerical results obtained with this method, in dimension 2 and 3.