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A particle-partition of unity method. VI: A \(p\)-robust multilevel solver. (English) Zbl 1065.65138

Griebel, Michael (ed.) et al., Meshfree methods for partial differential equations II. Selected contributions of the second international workshop, Bonn, Germany, September 15–17, 2003. Berlin: Springer (ISBN 3-540-23026-2/pbk). Lecture Notes in Computational Science and Engineering 43, 71-92 (2005).
Summary: We focus on the efficient multilevel solution of linear systems arising from a higher-order discretization of a second order partial differential equation using a partition of unity method. We present a multilevel solver which employs a tree-based spatial multilevel sequence in conjunction with a domain decomposition type smoothing scheme. The smoother is based on an overlapping subspace splitting, where the subspaces contain all interacting local polynomials.
The resulting local subspace problems are solved exactly. This leads to a computational complexity of the order \(O(Np^{3d})\) per iteration. The results of our numerical experiments indicate that the convergence rate of this multilevel solver is independent of the number of points \(N\) and the approximation order \(p\). Hence, the overall complexity of the solver is of the order \(O(\log(1/ \varepsilon)Np^{3d})\) to reduce the initial error by a prescribed factor \(\varepsilon\).
[For part V see M. Griebel and M. Schweitzer, Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations. Berlin: Springer, 519–542 (2003; Zbl 1033.65102).]
For the entire collection see [Zbl 1055.65003].

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms

Citations:

Zbl 1033.65102
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