A Systematic Approach to Constructing Preconditioners for the $hp$-Version Mass Matrix on Unstructured and Hybrid Finite Element Meshes

Abstract

We develop a systematic approach to building uniform preconditioners for the mass matrix on unstructured meshes composed of higher dimensional elements using only preconditioners for lower dimensional simplices. In particular, we show that the resulting preconditioners are automatically uniform with respect to both the mesh size h and polynomial degree $p$, and that the preconditioners can be implemented efficiently by exploiting the structure of the preconditioners on the lower dimensional elements. We illustrate the approach by developing preconditioners for prismatic elements and for the challenging case of hybrid meshes of hexahedra, prisms, and tetrahedra.