Assessing Curl-Conforming Bases for Pyramid Cells

Abstract

Successful three-dimensional finite element codes for Maxwell's equations must include and deal with all four types of geometrical shapes: tetrahedra, bricks, prisms, and quadrangular-based pyramids. However, pyramidal elements have so far been used very rarely because the basis functions associated with them have complicated expression, are complex in derivation, and have never been comprehensively validated. We recently published a simpler procedure for constructing higher-order vector bases for pyramid elements, so here we fill a gap by discussing a whole set of test case results that not only validate our new curl-conforming bases for pyramids, but which enable validation of other codes that use pyramidal elements for finite element method applications. The solutions of the various test cases are obtained using either higher order elements or multipyramidal meshes or both. Furthermore, the results are always compared with the solutions obtained with classical tetrahedral meshes using higher order bases. This allows us to verify that purely pyramidal meshes and elements give numerical results of comparable accuracy to those obtained with multitetrahedral meshes that use elements of the same order, essentially requiring the same number of degrees of freedom. The various results provided here also show that higher order vector bases always guarantee a superior convergence of the numerical results as the number of degrees of freedom increases.