Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

Reduced Systems for Three-Dimensional Elliptic Equations with Variable Coefficients

We consider large sparse nonsymmetric linear systems arising from finite difference discretization of three-dimensional (3D) convection-diffusion equations with variable coefficients. We show that performing one step of cyclic reduction yields a system of equations which is well conditioned and for which fast convergence can be obtained. A certain block ordering strategy is applied, and analyti...

A Massively Parallel Algebraic Multigrid Preconditioner based on Aggregation for Elliptic Problems with Heterogeneous Coefficients

This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. A greedy algorithm for the aggregation combined with an appropriate strength-of-connection criterion makes it especially suited for solving heterogeneous elliptic problems. Using decoupled aggregation on each process with data agglomeration onto fewer processes on the coarse level, it weakly ...

Elliptic regularity and solvability for PDEs with Colombeau coefficients

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of G∞-regularity of...

Parallel Application of a Novel Domain Decomposition Preconditioner for the Stable Finite Element Solution of Three-Dimensional Convection-Dominated PDEs

Boundary Value Problems for Linear PDEs with Variable Coefficients

A new method is introduced for studying boundary value problems for a class of linear PDEs with variable coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs with constant coefficients. As illustrative examples the following boundary value problems are solved: (a) A Dirichlet and a Neumann problem on the half line for th...