Olaf Steinbach

Numerical Approximation Methods for Elliptic Boundary Value Problems

Boundary Value Problems.- Function Spaces.- Variational Methods.- Variational Formulations of Boundary Value Problems.- Fundamental Solutions.- Boundary Integral Operators.- Boundary Integral Equations.- Approximation Methods.- Finite Elements.- Boundary Elements.- Finite Element Methods.- Boundary Element Methods.- Iterative Solution Methods.- Fast Boundary Element Methods.- Domain Decomposition Methods.

The construction of some efficient preconditioners in the boundary element method

The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally equivalent approximation.

Boundary Element Tearing and Interconnecting Methods

AbstractIn this paper we introduce the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. In some practical important applications such as far field computations, handling of singularities and moving parts etc., BETI methods have certainly some advantages over their finite element counterparts. This claim is especially true for the sparse versions of the BETI preconditioners resp. methods. Moreover, there is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. The first numerical results confirm the efficiency and the robustness predicted by our analysis.

On the stability of the L 2 projection in H 1 (Ω)

We prove the stability in H1(Ω) of the L2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the L2 projection in H1(Ω) holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.

Stability estimates for hybrid coupled domain decomposition methods

1 Preliminaries 1.1 Sobolev Spaces 1.2 Saddle Point Problems 1.3 Finite Element Spaces 1.4 Projection Operators 1.5 Quasi Interpolation Operators 2 Stability Results 2.1 Piecewise Linear Elements 2.2 Dual Finite Element Spaces 2.3 Higher Order Finite Element Spaces 2.4 Biorthogonal Basis Functions 3 The Dirichlet-Neumann Map for Elliptic Problems 3.1 The Steklov-Poincare Operator 3.2 The Newton Potential 3.3 Approximation by Finite Element Methods 3.4 Approximation by Boundary Element Methods 4 Mixed Discretization Schemes 4.1 Variational Methods with Approximate Steklov-Poincare Operators 4.2 Lagrange Multiplier Methods 5 Hybrid Coupled Domain Decomposition Methods 5.1 Dirichlet Domain Decomposition Methods 5.2 A Two-Level Method 5.3 Three-Field Methods 5.4 Neumann Domain Decomposition Methods 5.5 Numerical Results 5.6 Concluding Remarks References

Domain decomposition methods via boundary integral equations

Domain decomposition methods are designed to deal with coupled or transmission problems for partial dierential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of dierent discretization schemes. In general, the coupled problem is reduced to the Schur complement equation on the skeleton of the domain decomposition. Boundary integral equations are used to describe the local Steklov{Poincar e operators which are basic for the local Dirichlet{Neumann maps. Using dierent representations of the Steklov{Poincar e operators we formulate and analyze various boundary element methods employed in local discretization schemes. We give sucient conditions for the global stability and derive corresponding a priori error estimates. For the solution of the resulting linear systems we describe appropriate iterative solution strategies using both local and global preconditioning techniques.

Inexact Data-Sparse Boundary Element Tearing and Interconnecting Methods

The boundary element tearing and interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well-established finite element tearing and interconnecting (FETI) methods. In this paper we present inexact data-sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve twofold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps.

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

Abstract:In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results.

Coupled Boundary and Finite Element Tearing and Interconnecting Methods

We have recently introduced the Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established Finite Element Tearing and Interconnecting (FETI) methods. Since Finite Element Methods (FEM) and Boundary Element Methods (BEM) have certain complementary properties, it is sometimes very useful to couple these discretization techniques and to benefit from both worlds. Combining our BETI techniques with the FETI methods gives new, quite attractive tearing and interconnecting parallel solvers for large scale coupled boundary and finite element equations. There is an unified framework for coupling, handling, and analyzing both methods. In particular, the FETI methods can benefit from preconditioning components constructed by boundary element techniques. This is especially true for sparse versions of the boundary element method such as the fast multipole method which avoid fully populated matrices arising in classical boundary element methods.

The all-floating boundary element tearing and interconnecting method

Abstract The all-floating Boundary Element Tearing and Interconnecting method incooperates the Dirichlet boundary conditions by additional constraints in the dual formulation of the standard Tearing and Interconnecting methods. This simplifies the implementation, as all subdomains are considered as floating subdomains. The method shows an improved asymptotic complexity compared to the standard BETI approach. The all-floating BETI method is presented for linear elasticity in this paper.