A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations

Abstract

Time-domain finite-element solutions of Maxwell's equations require the solution of a sparse linear system involving the mass matrix at every time step. This process represents the bulk of the computational effort in time-dependent simulations. As such, mass lumping techniques in which the mass matrix is reduced to a diagonal or block-diagonal matrix are very desirable. In this paper, we present a special set of high order 1-form (also known as curl-conforming) basis functions and reduced order integration rules that, together, allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix. The method is derived from the Nedelec curl-conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite-element solutions to the second-order wave equation for the electric (or magnetic) field intensity. We present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments.