Numerical Approximation Methods for Elliptic Boundary Value Problems

Math Methods in App Sciences

2010

Self Cite

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some approriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.

“…As for the Laplace single layer potential, see for example [10,16], we conclude that the boundary integral operator A κ :…”

Section: Application Of Newton's Methodsmentioning

confidence: 55%

“…is linear and bounded, see, e.g., [10,15,16]. To normalize the eigensolutions t ∈ H −1/2 (Γ) of (2.1) we introduce a scaling condition by using an equivalent norm in H −1/2 (Γ),…”

Section: Application Of Newton's Methodsmentioning

confidence: 99%

“…Since the Fréchet derivative F ′ (t n , κ n ) is injective and satisfies a Gårdings inequality, the proof follows by applying standard arguments, see, e.g., [16].…”

Section: A Boundary Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Boundary element methods for Dirichlet boundary control problems

Phan

Math Methods in App Sciences

2010

Self Cite

“…For the numerical analysis of standard boundary element methods see, for example, Steinbach [2007]. Since the discretization of non-local boundary integral operators with singular kernel functions leads to dense stiffness matrices, the use of fast boundary element methods is an issue.…”

Section: Discussionmentioning

confidence: 99%

Challenges and Applications of Boundary Element Domain Decomposition Methods

Lecture Notes in Computational Science and Engineering

2008

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Boundary integral equation methods are well suited to represent the Dirichlet to Neumann maps which are required in the formulation of domain decomposition methods. Based on the symmetric representation of the local Steklov-Poincaré operators by a symmetric Galerkin boundary element method, we describe a stabilised variational formulation for the local Dirichlet to Neumann map. By a strong coupling of the Neumann data across the interfaces, we obtain a mixed variational formulation. For biorthogonal basis functions the resulting system is equivalent to nonredundant finite and boundary element tearing and interconnecting methods. We will also address several open questions, ideas and challenging tasks in the numerical analysis of boundary element domain decomposition methods, in the implementation of those algorithms, and their applications.

Boundary Element Methods: Foundation and Error Analysis

Hsiao

Encyclopedia of Computational Mechanics Second Edition

Wendland

2017

This chapter gives an expository introduction to the Galerkin‐BEM for the elliptic boundary value problems from the mathematical point of view. Emphasis will be placed on the variational formulations of the boundary integral equations (BIEs) and the general error estimates for the approximate solution in appropriate Sobolev spaces. A classification of BIEs will be given on the basis of the Sobolev index. The simple relations between the variational formulations of the BIEs and the corresponding partial differential equations under consideration will be indicated. Basic concepts such as stability, consistency, and convergence as well as the condition numbers and ill‐posedness, will be discussed by using elementary examples. This is the second edition of this chapter. The first edition was published in 2004. In this edition, additional topics have been included and discussed. These additional topics are concerned with the BIE‐method for a higher‐order elliptic equation, trace theorem and boundary spaces as well as numerical experiments for standard and fast multipole BEMs.