A Hodge Decomposition Finite Element Method for an Elliptic Maxwell Boundary Value Problem on General Polyhedral Domains

Abstract We develop a finite element method for an elliptic Maxwell boundary value problem on polyhedral domains in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {\mathbb{R}^{3}} with a general topology. Our method is based on a Hodge decomposition approach that leads to standard scalar elliptic problems and elliptic saddle point problems for vector potentials that have previously been investigated in the study of fluid flow problems. We carry out an error analysis that does not involve assumed regularity of the solution and present corroborating numerical results.