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\(hp\)-version additive Schwarz algorithms on triangular meshes. (English) Zbl 1038.65134

Domain decomposition Dirichlet-Dirichlet solvers for \(hp\)-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. Edge coordinate functions are allowed to be either nodal, with special choices of nodes, or hierarchical polynomials of several types.
Two kinds of interior coordinate functions are studied: arbitrary and so-called discrete quasi-harmonic functions. In all situations, preconditioners are suggested which are (almost) spectrally equivalent to the global stiffness matrix, which require only element-by-element and edge-by edge operations, thus providing a high level of parallelization. They also considerably reduce the computational cost.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
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