The Characterization of Boundary Integral Operators and Galerkin Boundary Element Methods

1990 ◽  
pp. 239-255
Author(s):  
Albert H. Schatz ◽  
Vidar Thomée ◽  
Wolfgang L. Wendland
Keyword(s):  
Boundary Element ◽  
Integral Operators ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Boundary Integral Operators ◽  
Galerkin Boundary Element Methods

scholarly journals Adaptive boundary-element methods for transmission problems

1997 ◽  
Vol 38 (3) ◽  
pp. 336-367 ◽  
Author(s):  
Carsten Carstensen ◽  
Ernst P. Stephan
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Integral Operators ◽  
Adaptive Algorithms ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
A Posteriori ◽  
Multiplicative Constant ◽  
Transmission Problems ◽  
Weakly Singular

AbstractIn this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

scholarly journals The Boundary Element Method in Acoustics: A Survey

2019 ◽  
Vol 9 (8) ◽  
pp. 1642 ◽  
Author(s):  
Kirkup
Keyword(s):  
Integral Equation ◽  
Computational Complexity ◽  
Boundary Element Method ◽  
Inverse Problems ◽  
Boundary Element ◽  
Integral Operators ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Software Library ◽  
Element Method

The boundary element method (BEM) in the context of acoustics or Helmholtz problems is reviewed in this paper. The basis of the BEM is initially developed for Laplace’s equation. The boundary integral equation formulations for the standard interior and exterior acoustic problems are stated and the boundary element methods are derived through collocation. It is shown how interior modal analysis can be carried out via the boundary element method. Further extensions in the BEM in acoustics are also reviewed, including half-space problems and modelling the acoustic field surrounding thin screens. Current research in linking the boundary element method to other methods in order to solve coupled vibro-acoustic and aero-acoustic problems and methods for solving inverse problems via the BEM are surveyed. Applications of the BEM in each area of acoustics are referenced. The computational complexity of the problem is considered and methods for improving its general efficiency are reviewed. The significant maintenance issues of the standard exterior acoustic solution are considered, in particular the weighting parameter in combined formulations such as Burton and Miller’s equation. The commonality of the integral operators across formulations and hence the potential for development of a software library approach is emphasised.

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