The finite element method (FEM) is an extremely successful approach to numerically approximating solutions to partial differential equations arising from many real-world scenarios. While the p- and hp-versions offer significant advantages over the ordinary h-version method, poor conditioning of the elemental matrices may neutralize many of those improvements. The construction of efficient, domain decomposition type preconditioners for the p-version mass matrix is thus of practical interest, particularly when one turns to applications beyond Poisson-type problems. We consider the problem of preconditioning the mass matrix arising from high order FEM methods on a variety of elements, including triangles, tetrahedra and tensor product elements. In all cases, a non-overlapping, Additive Schwarz Method is presented with a preconditioned system for which the condition number is bounded independently of the polynomial order and the mesh size. Furthermore, we also construct efficient preconditioners for any linear combination of mass and stiffness matrix on triangles, and present algorithms for the efficient implementation of the preconditioner on triangles using Bernstein polynomials.
Jiang, Shuai,
"Preconditioning the p-FEM Mass Matrix: Theory, Implementation, and Applications"
(2020).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.26300/e75a-f467
