Robust multilevel methods for quadratic finite element anisotropic elliptic problems |
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Authors: | Johannes Kraus Maria Lymbery Svetozar Margenov |
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Affiliation: | 1. Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, , Altenberger Strasse?69, A‐4040 Linz, Austria;2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, , Acad. G. Bonchev Street, Block?25A, 1113 Sofia, Bulgaria |
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Abstract: | This paper discusses a class of multilevel preconditioners based on approximate block factorization for conforming finite element methods employing quadratic trial and test functions. The main focus is on diffusion problems governed by a scalar elliptic partial differential equation with a strongly anisotropic coefficient tensor. The proposed method provides a high robustness with respect to non‐grid‐aligned anisotropy, which is achieved by the interaction of the following components: (i) an additive Schur complement approximation to construct the coarse‐grid operator; (ii) a global block (Jacobi or Gauss–Seidel) smoother complementing the coarse‐grid correction based on (i); and (iii) utilization of an augmented coarse grid, which enhances the efficiency of the interplay between (i) and (ii). The performed analysis indicates the high robustness of the resulting two‐level method. Moreover, numerical tests with a nonlinear algebraic multilevel iteration method demonstrate that the presented two‐level method can be applied successfully in the recursive construction of uniform multilevel preconditioners of optimal or nearly optimal order of computational complexity. Copyright © 2013 John Wiley & Sons, Ltd. |
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Keywords: | anisotropic elliptic problems robust preconditioning quadratic finite elements multilevel methods additive Schur complement approximation overlapping domain decomposition |
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