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Article
Higher-order time domain boundary elements for elastodynamics: graded meshes and hp versions
The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-op...
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Chapter
Diagonal Scaling Preconditioner and Locally-Refined Meshes
In this chapter, we study the effect of mesh refinements on the condition numbers of the stiffness matrices arising from the h-version Galerkin boundary element discretisation of the weakly-singular and hypersing...
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Chapter
Two-Level Methods: The p-Version
Similarly to Chapter 3, this chapter deals with the same issue for the p-version.
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Chapter
Additive Schwarz Methods for the hp-Version
This chapter deals with additive Schwarz methods for the hp-version on quasi-uniform and graded meshes. Multilevel methods are also analysed.
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Chapter
Indefinite Problems
This chapter designs and analyses additive Schwarz methods for boundary integral equations arising from the Helmholtz equation in two dimensions. A general theory for indefinite problems is developed and appli...
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Chapter
Multilevel Preconditioners with Adaptive Mesh Refinements
In this chapter we study the use of multilevel methods to obtain optimal preconditioners in the case that the mesh is locally refined. Local mesh refinements, particularly important for boundary element discre...
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Chapter
Introduction
This chapter sets the scene for the problems to be studied in this book. Typical problems from which boundary integral equations arise, namely, the scattering of time harmonic waves and cracks in elastic media...
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Chapter
Two-Level Methods: The h-Version
This chapter designs and analyses non-overlap** and overlap** additive and multiplicative Schwarz methods for the h-version. The novelty of this chapter is that in order to prove the coercivity of the decompo...
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Chapter
Multilevel Methods: The h-Version
When applying the two-level methods for the h-version, we have to solve a global problem on a coarse space. However, the dimensions of this coarse space can be large. Therefore it is natural to carry out the same...
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Chapter
A Fully Discrete Method
In this chapter, non-overlap** and overlap** additive Schwarz and multilevel preconditioners are analysed for a fully-discrete and symmetric method for the weakly singular integral equation
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Chapter
Implementation Issues and Numerical Experiments
In this chapter we discuss issues with implementation for the two-dimensional problems discussed in this part of the book. First we mention some important issues in implementation of the methods and then prese...
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Chapter
Two-Level Methods: The hp-Version on Rectangular Elements
IIn this chapter, we study different two-level methods for the hp-version of the Galerkin method with rectangular elements. Both the hypersingular and weakly-singular integral equations on a screen in three dimen...
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Chapter
FEM–BEM Coupling
The coupling of boundary elements and finite elements is a powerful tool for the numerical treatment of boundary value problems on unbounded domains, and in particular, of interface problems. It allows us to b...
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Chapter
General Framework of Preconditioners
This chapter provides the general framework of preconditioning by additive and multiplicative Schwarz operators. Preconditioners are viewed from different angles: in the matrix form, variational form, and oper...
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Chapter
Two-Level Methods: The Hp-Version on Triangular Elements
In this chapter we report on the paper [111] by Heuer, Leydecker, and Stephan, which analyses a two-level additive Schwarz method for the hp-version on triangular elements applied to the hypersingular integral eq...
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Article
A residual a posteriori error estimate for the time–domain boundary element method
This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and a...
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Article
Collocation with trigonometric polynomials for integral equations to the mixed boundary value problem
We consider the direct boundary integral equation formulation for the mixed Dirichlet–Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system o...
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Article
Boundary elements with mesh refinements for the wave equation
The solution of the wave equation in a polyhedral domain in \({\mathbb {R}}^3\) ...
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Chapter
A Discrete Collocation Method for a Hypersingular Integral Equation on Curves with Corners
This paper is devoted to the approximate solution of a hypersingular integral equation on a closed polygonal boundary in ...
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Article
Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations
For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. There...
