A dual finite element complex on the barycentric refinement

Abstract

Abstract. Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex X • centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex Y • of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the L 2 duality is non-degenerate on Y i × X 2−i for each i ∈ {0, 1, 2}. In particular Y 1 is a space of curl-conforming vector fields which is L 2 dual to Raviart-Thomas div-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

Low frequency scaling of the mixed MFIE for scatterers with a non-simply connected surface

“…Two sets of functions will be used in the rest of this paper: the familiar RWG functions [2] and the Buffa-Christiansen (BC) or ChenWilton functions [4,5]. Both these sets of functions have one function per edge of the mesh.…”

“…Instead of using the familiar RWG functions for the testing, this scheme uses the rotated Buffa-Christiansen or Chen-Wilton functions [4,5]. In [1], it was shown that the solution current of this so-called mixed MFIE is lowfrequency stable, which is in stark contrast with the standard MFIE.…”

A well-conditioned combined field integral equation based on quasi-helmholtz projectors

“…are the EFIE operators discretized with standard RWG f [9] and BC basis functions f BC [10] respectively, Kk…”

On the Direct Evaluation of Surface Integral Equation Impedance Matrix Elements Involving Point Singularities

“…The above-mentioned integrals can be categorized into weakly singular (1) and (2) and strongly singular (3), respectively. Moreover, one can easily replace RWG functions with the novel Buffa-Cristiansen functions [10] without affecting the overall mathematical complexity since the latter are linear combinations of the former on the barycentric mesh. Finally, smart usage of both functions can lead to well-tested SIE operators, improving the overall accuracy of SIE solvers, provided of course that the associated impedance matrix elements are accurately computed.…”