Improvements for some condition number estimates for preconditioned system in <i>p</i>‐FEM

Beuchler, Sven; Braess, Dietrich

doi:10.1002/nla.489

Cited by 4 publications

(4 citation statements)

References 20 publications

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“…We will conclude the paper with a remark about an application for the p-version of the FEM in three dimensions. Using the basis of the integrated Legendre polynomials {L i } p i=2 , it has been proved in [3] that the element stiffness matrix for odd polynomial degree p with respect to the interior bubbles is spectrally equivalent to the matrix…”

Section: The General Casementioning

confidence: 99%

Overlapping Additive Schwarz Preconditioners for Elliptic Problems with Degenerate Locally Anisotropic Coefficients

Beuchler¹,

Nepomnyaschikh²

2007

SIAM J. Numer. Anal.

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In this paper, we consider degenerate and locally anisotropic boundary value problems on the unit square. These problems are discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned gradient method using a domain decomposition preconditioner with overlap. We prove that the condition number of the preconditioned system is bounded by a constant independent of the discretization parameter. Moreover, the preconditioning operation requires O(N ) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.

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Section: The General Casementioning

confidence: 99%

Overlapping Additive Schwarz Preconditioners for Elliptic Problems with Degenerate Locally Anisotropic Coefficients

Beuchler¹,

Nepomnyaschikh²

2007

SIAM J. Numer. Anal.

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“…It was designed as a solver for the preconditioner, which is the spectrally equivalent finite element version of the finite-difference preconditioner of [Ivanov and Korneev (1996)]. Due to one simplification of the finite-difference/finite element preconditioner, independently approved by [Korneev et al (2003b)], [Beuchler et al (2004)], and [Beuchler and Braess (2006)], the cost of the multigrid preconditioner-solver was reduced to the optimal order, i.e. O(p 2 ).…”

Section: Fast Dirichlet Solvers For 2d Reference Elementsmentioning

confidence: 99%

Dirichlet–Dirichlet Domain Decomposition Methods for Elliptic Problems

Korneev

Langer

2013

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PrefaceDomain decomposition (DD) methods provide powerful tools for constructing parallel numerical solution algorithms for large scale systems of algebraic equations arising from the discretization of partial differential equations. The Alternating Schwarz Method, proposed by H. Schwarz (1869) in order to prove the existence of harmonic functions with prescribed Dirichlet data on the boundary of complicated domains, and substructuring techniques, developed by engineers in the 60s of the preceding century in order to create faster procedures for the analysis of complex structures, are commonly accepted as the origins of modern DD methods. Thus, DD methods have been developed for a long time, but most extensively since the first international DD conference that was held at Paris in 1987. This concerns both the theory and the practical use of DD techniques for creating efficient application software for massive parallel computers. The advances in DD are well documented in the proceedings of the international DD conferences 1 since 1987 and numerous papers. The first textbook on DD methods was published by B.F. Smith, P.E. Bjørstad, and W. Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic ProblemsPechstein has recently published a monograph on Finite Element Tearing and Interconnecting (FETI) and Boundary Element Tearing and Interconnecting (BETI) methods for a special class of multiscale problems. Our book is different from the textbooks mentioned above and very different from Pechstein's monograph. We mainly discuss one special class of primal substructuring methods also called Dirichlet-Dirichlet DD methods, and we emphasize the peculiarities of their application to hp finite element equations. In particular, we discuss and analyze the inexact versions of these DD methods which lead to optimal or, at least, almost optimal complexity and high efficiency in practical applications. The optimization of several important components of DD algorithms for hp discretization was achieved only quite recently. These topics are not discussed or at least not sufficiently discussed in the books mentioned above. The contributions of the authors to this field mainly appeared in journal publications including joint papers. The reader will become familiar with inexact Dirichlet-Dirichlet DD methods enjoying optimal complexity and the techniques for the numerical analysis of these methods. Thus, we hope that readers interested in both practice and theory can benefit from our book. In particular, the book will open the possibility for the reader to use such DD methods in new fields of applications like multiphysics applications in Computational Sciences. In this book, the reader can find a brief historical overview, the basic results of the general theory of domain and space decompositions as well as the description and analysis of practical DD algorithms. Elliptic problems with strongly jumping coefficients are daily met in the engineering practice. Numerical techniques should not deteriorate in efficiency when solvin...

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“…Based on the theoretical estimates in [24], [25], the stability proof has been obtained for the decomposition of the Schur complement related to the element boundaries into the Schur complement related to the face bubbles on each face separately and the wire-basket corresponding to vertex and edge bubble functions. In [6], see also [5] for the refined estimates, a first solver for the interior bubbles is developed in the basis of the integrated Legendre polynomials (1.3). This solver uses a basis [Ψ]…”

Section: Introductionmentioning

confidence: 99%

“…The main goal is the development of another p-wavelet basis with properties as in (1.4) for spaces of the type B p,1 = {u ∈ P p (−1, 1), u(1) = 0}. This is an extension of the results presented in [6], [5]. Moreover, the solvers use the tensor product structure of the elements and patches directly.…”

Section: Introductionmentioning

confidence: 99%

Wavelet solvers for hp-FEM discretizations in 3D using hexahedral elements

Beuchler

2009

Computer Methods in Applied Mechanics and Engineering

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In this paper we investigate the discretization of an elliptic boundary value problem in 3D by means of the hp-version of the finite element method using a mesh of hexahedrons. The corresponding linear system is solved by a preconditioned conjugate gradient method with an overlapping preconditioner as inexact additive Schwarz preconditioner. The remaining subproblems are treated by a tensor product based preconditioner. This preconditioner uses a basis transformation into a basis which is stable in L2 and H 1. Several numerical examples show the efficiency of the proposed method.

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Improvements for some condition number estimates for preconditioned system in p‐FEM

Cited by 4 publications

References 20 publications

Overlapping Additive Schwarz Preconditioners for Elliptic Problems with Degenerate Locally Anisotropic Coefficients

Overlapping Additive Schwarz Preconditioners for Elliptic Problems with Degenerate Locally Anisotropic Coefficients

Dirichlet–Dirichlet Domain Decomposition Methods for Elliptic Problems

Wavelet solvers for hp-FEM discretizations in 3D using hexahedral elements

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