A note on the fast direct method for discrete elliptic problems
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Sergei A. Goreinov
Abstract
We consider a method due to P. Vassilevski and Yu. A. Kuznetsov [4, 10] for solving linear systems with matrices of low Kronecker rank such that all factors in Kronecker products are banded. Most important examples of such matrices arise from discretized div K grad operator with diffusion term k1(x)k2(y)k3(z). Several practical issues are addressed: an MPI implementation with distribution of data along processor grid inheriting Cartesian 3D structure of discretized problem; implicit deflation of the known nullspace of the system matrix; links with two-grid framework of multigrid algorithm which allow one to remove the requirement of Kronecker structure in one or two of axes. Numerical experiments show the efficiency of 3D data distribution having the scalability analogous to (structured) HYPRE solvers yet the absolute timings being an order of magnitude lower, on the range from 10 to 104 cores.
Funding: The work was supported by the world-class research center ‘Moscow Center for Fundamental and Applied Mathematics’.
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- Numerical model for the investigation of hydrodynamic stability of shear flows in pipes of elliptic cross-section
- A note on the fast direct method for discrete elliptic problems
- Simulation of a random field with given distribution of one-dimensional integral
- A new algorithm for numerical modelling of impurity transport in the frame of statistically homogeneous sharply contrasting media
- Approximate spectral models of random processes with periodic properties
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