Comparison of finite element and discrete exterior calculus in computation of time-harmonic wave propagation with controllability

Abstract

This paper discusses computation of time-harmonic wave problems using a mixed formulation and the controllability method introduced by Roland Glowinski. As an example, a scattering problem (in an exterior domain) is considered, and the continuous problem is first formulated in terms of differential forms. Based on the continuous formulation, we write the discrete problem and the controllability algorithm for methods based on both the finite element exterior calculus (FEEC) and the discrete exterior calculus (DEC). As the discrete exterior calculus method provides us with a diagonal “mass matrix”, time-stepping in the DEC approach is remarkably more efficient than in the FEEC approach. For the computations in this paper, we choose the lowest order Whitney elements (a.k.a. Raviart–Thomas elements) for the FEEC approach, and in the DEC discretization we use different diagonal approximations for the Hodge star. Especially, in the DEC approach, a “harmonic Hodge” approximation is used, the derivation of which is based on the time-harmonicity of the problem. Different type of grids are used to study the sensitivity of the solution to the quality of the grid. Putting an effort on meshes regular enough, the computed DEC-solution is as accurate as the FEEC-solution, but reached in the fraction of the time. Both methods seem to be able to keep the solution accuracy rather well in computations with a high wave number (corresponding to a high frequency and a small wave length).