An iterated tangential filtering decomposition. (English) Zbl 1071.65035

The authors consider solving large linear algebraic systems with symmetric and positive definite matrices and a tridiagonal block structure arising from the discretization of elliptic partial differential equations on structured meshes mainly in the tridimensional space. The described techniques can be extended to nonsymmetric case.
The topic of this paper consists on construction of a block LDL\(^{\text{T}}\) preconditioner having the same block LDL\(^{\text{T}}\) structure as the original system. Various tangential filtering techniques are discussed in the first part of the paper. The preconditioners are used in the preconditioned conjugate gradient algorithm (PCG) combined with the preconditioned Richardson method. The amount of memory needed by the preconditioner in a factorized form is smaller than that needed by the original matrix. A C++ implementation is outlined. The paper contains wide numerical tests. Elliptic partial differential equations in a given domain in \(\mathbb R^3\) are discretized on uniform, non-uniform and very anisotropic grids. Highly heterogeneous media are, moreover, considered. The PCG method is used and the results are compared for various input data.
Reviewer: Jan Zítko (Praha)


65F10 Iterative numerical methods for linear systems
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
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