A Numerical Comparison of an Isogeometric and a Parametric Higher Order Raviart–Thomas Approach to the Electric Field Integral Equation

Abstract

In this paper, we advocate a novel spline-based isogeometric approach for boundary elements and its efficient implementation. We compare solutions obtained by both an isogeometric approach, and a classical parametric higher-order approach via Raviart-Thomas elements to the solution of the electric field integral equation; i.e., the solution to an electromagnetic scattering problem, promising high convergence orders w.r.t. pointwise error. We discuss both, the obtained accuracy per DOF, as well as the effort required to solve the corresponding system iteratively, on three numerical examples of varying complexity.

Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

“…As in Dölz et al, 30 we stress that this is an optimistic estimate to the accuracy of Raviart‐Thomas spaces, as the geometry representation through the parametric mapping is still exact. In a traditional, mesh‐based RT‐approach, an additional error would be introduced through an approximation of the geometry through a mesh.…”

“…In its discrete form, this problem induces a linear system which we choose to assemble as in Dölz et al 11,30 Therein, can be interpreted as a matrix valued function dependent on κ h .…”

“…the same underlying “mesh.” For a definition of these quadrilateral Raviart‐Thomas elements, we refer to Zaglmayr 45 . For a detailed explanation of the experimental setup, see Dölz et al, 30 where the same comparison has been conducted for electromagnetic scattering problems.…”

Analysis and Implementation of Isogeometric Boundary Elements for Electromagnetism

“…A comparison of the isogeometric ansatz spaces to different discretisations of the electric field integral equation was first conducted by us in [47,75], where we compare the spline-based discretisation to parametric Raviart-Thomas elements and also discuss spectral elements, respectively. Our implementation used for the numerical studies of [47,50,75] was made publicly available [44], archived [43], and its API is explained in [45]. To the best of the authors' knowledge, this is the only publicly available isogeometric boundary element code, and the only publicly available code for the EFIE capable of higher-order computations.…”

“…In addition to the visualisations, detailed numerical data of selected simulations are depicted in Table 5.3. All data and some figures presented within this section were first published in [47] GMRES iterations max pw. error p = 1 B-spline p = 1 RT p = 2 B-spline p = 2 RT p = 3 B-spline p = 3 RT p = 4 B-spline p = 4 RT…”