Jason Kurtz

Sparse direct factorizations through unassembled hyper-matrices

Abstract We present a novel strategy for sparse direct factorizations that is geared towards the matrices that arise from hp -adaptive Finite Element Methods. In that context, a sequence of linear systems derived by successive local refinement of the problem domain needs to be solved. Thus, there is an opportunity for a factorization strategy that proceeds by updating (and possibly downdating) the factorization. Our scheme consists of storing the matrix as unassembled element matrices, hierarchically ordered to mirror the refinement history of the domain. The factorization of such an ‘unassembled hyper-matrix’ proceeds in terms of element matrices, only assembling nodes when they need to be eliminated. The main benefits are efficiency from the fact that only updates to the factorization are made, high scalar efficiency since the factorization process uses dense matrices throughout, and a workflow that integrates naturally with the application.

hp-Adaptive Finite Elements for Coupled Multiphysics Wave Propagation Problems

The paper describes a generalization of hp-adaptive finite elements technology to coupled multiphysics problems. Three representative examples are used: dual-mixed formulation with weakly imposed symmetry for linear elasticity, coupled acoustics/viscoelasticity and coupled acoustics/poroelasticity problem, to illustrate variational formulations and concept of weak couplings. We discuss then necessary changes in data structures, constrained approximation and the hp mesh optimization algorithm. Sample numerical results for the three problems illustrate the new methodology.

Sparse direct factorizations through unassembled hyper‐matrices

We set out to efficiently compute the solution of a sequence of linear systems Aixi = bi, where the matrix Ai is tightly related to matrix Ai –1. In the setting of an hp -adaptive Finite Element Method, the sequence of matrices Ai results from successive local refinements of the problem domain. At any step i > 1, a factorization already exists and it is the updated linear system relative to the refined mesh for which a factorization must be computed in the least amount of time. This observation holds the promise of a tremendous reduction in the cost of an individual refinement step. We argue that traditional matrix storage schemes, whether dense or sparse, are a bottleneck, limiting the potential efficiency of the solvers. We propose a new hierarchical data structure, the Unassembled Hyper-Matrix (UHM), which allows the matrix to be stored as a tree of unassembled element matrices, hierarchically ordered to mirror the refinement history of the physical domain. The factorization of such an UHM proceeds in terms of element matrices, only assembling nodes when they need to be eliminated. Efficiency comes in terms of both performance and space requirements. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Conjugated Bubnov-Galerkin Infinite Element for Maxwell Equations

We propose a (conjugated) Bubnov-Galerkin Infinite Element (IE) discretization for the time-harmonic Maxwell scattering and radiation problems. The element falls into a family of infinite elements satisfying an exact sequence property. The exact sequence results from incorporating the far-field pattern into the anzatz for the solution and the test functions, and it differs from the standard grad-curl-div sequence. We verify the construction with 2D numerical experiments.