An iterative substructuring method for the system of linear elasticity in three dimensions is introduced and analyzed. The pure displacement formulation for compressible materials is discretized with the spectral element method. The resulting stiiness matrix is symmetric and positive deenite. The method proposed provides a domain decomposition preconditioner constructed from local solvers for t...

This paper describes an efficient and robust hybrid parallel solver ‘‘the SPIKE algorithm’’ for narrow-banded linear systems. Two versions of SPIKE with their built-in-options are described in detail: the Recursive SPIKE version for handling non-diagonally dominant systems and the Truncated SPIKE version for diagonally dominant ones. These SPIKE schemes can be used either as direct solvers, or ...

We present hybrid finite element methods, which are equivalent to a discontinuous Galerkin method based on the ultra weak variational formulation (UWVF) by Cessenat and Despres. When solving a scalar or vectorial wave equation with hybrid finite elements, normal and tangential continuity of the flux field, respectively, is broken across element interfaces and reinforced again by introducing hyb...

The popularity of direct methods for solving large sparse linear systems of equations Ax = b stems from their generality and robustness. Indeed, if numerical pivoting is properly incorporated, direct solvers rarely fail for numerical reasons; the main reason for failure is a lack of memory. Although increasing amounts of main memory have enabled direct solvers to solve many problems that were p...

The elliptic Monge-Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community [Tad12, FGN13]. Fast three dimensional solvers are needed, for example in medical image registration, [HZTA04, HRT10, HPM+09, HM07], but are not yet available. We build fast solvers for smooth solutions in three dimensions...

Multigrid can be formulated as an algorithm for an abstract problem that is independent of the partial di erential equation, domain, discretization method, and number of processors. In such an abstract setting, problems not arising from partial di erential equations can be treated also (c.f. aggregation-disaggregation methods). Quite general theory exists for linear problems, e.g., C. C. Dougla...

We give new algorithms for computing basis cochains for real-valued homology, cohomology, and harmonic cochains on manifold simplicial complexes. We discuss only planar, surface, and solid meshes. Our algorithms are based on a least squares formulation. Previous methods for computing homology and cohomology have relied on persistence algorithm or Smith normal form, both of which have cubic comp...

In this paper a finite difference method for solving 2-dimensional diffusion equation is presented. The method employs Crank-Nicolson scheme to improve finite difference formulation and its convergence and stability. The obtained solution will be a recursive formula in each step of which a system of linear equations should be solved. Given the specific form of obtained matrices, rather than sol...