Interpolatory Curl-Conforming Pyramidal Elements: Progress and Results

Advanced finite element codes for the analysis and synthesis of three-dimensional electromagnetic structures should be able to use hybrid meshes composed of cells of different shapes, namely cuboids, tetrahedra, prisms and pyramids. For all these elements, with the exception of pyramids, it has long been known how to define divergence-conforming or curl-conforming higher-order basis functions, whether interpolatory or hierarchical [1]. We know that for all elements it is better to work in a parent space where a parent cell is defined. The cells of the observer domain (otherwise called child domain) are obtained using appropriate nodal shape functions which, like the vector basis functions, are polynomials of the parent variables. Unfortunately, things are not so simple for the pyramid because it has four edges converging on its tip. Within a geometrically continuous hybrid mesh, i.e. without gaps between cells, the tangential or normal continuity of the vector basis functions on the boundaries of the pyramids is guaranteed only by accepting that the shape functions and the basis functions of the pyramid are fractional functions with a tip singularity. This meant that for a long time it was not possible to properly define the order of the bases nor was it possible to demonstrate the completeness of the set of shape functions and vector bases of the pyramidal element. As demonstrated in [2, 3], shape functions and vector bases take polynomial form if defined in a grandparent space, different from the parent one. In the grandparent space the pyramid has the shape of a cube (see Fig. 1). Therefore, using grandparent variables we can clearly define the order of the bases (since we work with sets of polynomials) while ensuring the required continuity from element to element

On a Novel Calderón Preconditioning Strategy Based on High-Order Quasi-Helmholtz Projectors

The first of Calderón identities indicates that the operator of the electric field integral equation (EFIE) can potentially act as a preconditioner for itself. However, the discretization of this preconditioning operator via the boundary element method (BEM) requires constructing a dual space commonly defined on a barycentric refinement of the orginal mesh. This contribution introduces a novel Calderón strategy that leverages high-order quasi-Helmholtz projectors. Unlike the standard strategies, which are based on barycentric refinements, a dual space is obtained by combinating functions of higher order. This strategy allows the development of a discretization of the Calderón identity that is suitable for arbitrary order. The well-conditioning of the proposed formulation is investigated, and numerical results are presented to corroborate the theory

On Operator Filtering for Integral Equations: The High-Order Case

In this work, we extend the standard quasi-Helmholtz filters to high-order discretizations in the context of the boundary element method. This generalization allows preconditioning integral equations such as the electric field integral equation (EFIE) in the h-refinement regime while preserving the accuracy gain of high-order discretizations. The theoretical framework will be corroborated by numerical results that validate the effectiveness of the proposed strategy when stabilizing the refinement-dependent spectral behavior of the high-order discretized EFIE

Interpolatory Curl-Conforming Vector Bases for Pyramid Cells

We have recently shown that hierarchical higher-order complete curl-conforming and divergence-conforming bases for pyramids can be obtained by multiplying the lowest-order basis functions by hierarchical scalar multipliers defined by Jacobi polynomials. This paper extends this technique and builds curl-conforming interpolatory bases for pyramids by replacing the hierarchical polynomials with appropriate combinations of interpolatory polynomials of Silvester. Our curl-conforming bases for the pyramid are tangentially continuous with those of adjacent differently shaped cells of the same order and type (i.e., hierarchical or interpolatory) available for years in the literature. This allows numerical electromagnetic solvers using zero-order vector basis functions to be transformed into higher order solvers that work with hybrid meshes simply by adding a few routines to compute the multiplicative polynomials and their first derivatives. Hierarchical bases, including ours of previous papers, are in general more convenient than interpolatory ones for using p-adaptive techniques, while the interpolatory bases such as those shown here are more easily implemented because the recurrence relations of Silvester polynomials are much simpler than those associated with hierarchical multipliers. Numerical results that verify the correctness of our new bases are also reported