A Finite Element-Capacitance Matrix Method for the Neumann Problem for Laplace’s Equation

Proskurowski, Włodzimierz; Widlund, Olof B.

doi:10.1137/0901029

Cited by 37 publications

(21 citation statements)

References 26 publications

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“…A similar result has been established in [27,4] for algebraic systems of linear equations with symmetric coefficient matrices. Such methods are closely related to the capacitance matrix methods; see, e.g., [36,11]. Many domain decomposition methods can also be used for the effective solution of problems with rough coefficients and can provide uniform convergence.…”

Section: Andrew Knyazev and Olof Widlundmentioning

confidence: 99%

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Knyazev

Widlund

2001

Math. Comp.

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Abstract. We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

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Section: Andrew Knyazev and Olof Widlundmentioning

confidence: 99%

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Knyazev

Widlund

2001

Math. Comp.

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“…[BDGG], [BW], [OW], [PW1], [PW2]) ; une brève discussion de leurs principes peut être trouvée dans [At]. Ici, nous allons analyser l'approche que nous avons proposée et étudiée en détail dans [At] ; cet article est en fait la publication d'une partie de ce travail.…”

Section: Introduction Position Du Problèmeunclassified

Une analyse de la méthode des domaines fictifs pour le problème de Helmholtz extérieur

Atamian¹,

Joly²

1993

ESAIM: M2AN

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“…Hence, this capacitance matrix algorithm requires the computation of all the eigenvectors in the null spaces of A, A T , B, and B T . The capacitance matrix algorithm has also been employed as a discrete analogy to the potential theory for partial differential equations by Proskurowski and Wildlund [16,17] and others. However, the framework of their capacitance matrix algorithms is limited to solving second-order elliptic boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

Lai¹,

Vemuri²

1998

SIAM J. Sci. Comput.

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Abstract. The capacitance matrix method has been widely used as an efficient numerical tool for solving the boundary value problems on irregular regions. Initially, this method was based on the Sherman-Morrison-Woodbury formula, an expression for the inverse of the matrix (A + UV T ) with A ∈ n×n and U, V ∈ n×p . Extensions of this method reported in literature have made restrictive assumptions on the matrices A and (A + UV T ). In this paper, we present several theorems which are generalizations of the capacitance matrix theorem in [4] and are suited for very general matrices A and (A + UV T ). A generalized capacitance matrix algorithm is developed from these theorems and holds the promise of being applicable to more general problems; in addition, it gives ample freedom to choose the matrix A for developing very efficient numerical algorithms. We demonstrate the usefulness of our generalized capacitance matrix algorithm by applying it to efficiently solve large sparse linear systems in several computer vision problems, namely, the surface reconstruction and interpolation, and the shape from orientation problem.

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A Finite Element-Capacitance Matrix Method for the Neumann Problem for Laplace’s Equation

Cited by 37 publications

References 26 publications

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Une analyse de la méthode des domaines fictifs pour le problème de Helmholtz extérieur

Generalized Capacitance Matrix Theorems and Algorithm for Solving Linear Systems

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