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It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability condition: 's'sThe mesh width h of the finite element mesh has to satisfy k 2 h≲1'', where k denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions for large wave numbers k≳50. In our paper, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability of the discretisation can be checked through an 's'salmost invariance'' condition. As an application, we will consider a one-dimensional finite element space for the Helmholtz equation and apply our theory to prove stability under the weakened condition hk≲1 and optimal convergence estimates

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation (RPIE) on the bounded surface of the scatterer. We formulate an algorithm for the spacetime Galerkin discretization with smooth and compactly supported temporal basis functions which have been introduced in [S. Sauter and A. Veit: A Galerkin Method for Retarded Boundary Integral Equations with Smooth and Compactly Supported Temporal Basis Functions, Preprint 04-2011, Universitat Zurich]. For the debugging of an implementation and for systematic parameter tests it is essential to have some explicit representations and some analytic properties of the exact solutions for some special cases at hand. We will derive such explicit representations for the case that the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed method. AMS subject classifications. 35L05, 65R20

In this paper, we consider the numerical discretization of elliptic eigenvalue problems by Finite Element Methods and its solution by a multigrid method. From the general theory of finite element and multigrid methods, it is well known that the asymptotic convergence rates become visible only if the mesh width h is sufficiently small, h ≤ h 0. We investigate the dependence of the maximal mesh width h 0 on various problem parameters such as the size of the eigenvalue and its isolation distance. In a recent paper (Sauter in Finite elements for elliptic eigenvalue problems in the preasymptotic regime. Technical Report. Math. Inst., Univ. Zurich, 2007), the dependence of h 0 on these and other parameters has been investigated theoretically. The main focus of this paper is to perform systematic experimental studies to validate the sharpness of the theoretical estimates and to get more insights in the convergence of the eigenfunctions and -values in the preasymptotic regime.

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.

In this paper, we will present a new approach for solving boundary integral equations with panel clustering. In contrast to all former versions of panel clustering, the computational and storage complexity of the algorithm scales linearly with respect to the number of degrees of freedom without any additional logarithmic factors. The idea is to develop alternative formulations of all classical boundary integral operators for the Laplace problem where the kernel function has a reduced singular behaviour. It turns out that the application of the panel-clustering method with variable approximation order preserves the asymptotic convergence rate of the discretisation and has significantly reduced complexity.

In many cases, boundary value problems on a domain Ω can be rewritten as integral equations on the boundary of Ω. The discretization of this integral equation leads to a system of linear equations with a dense coefficient matrix of dimension N. In this paper, we will present the panel clustering algorithm which avoids the generation of the N 2 matrix entries by representing the integral operator on the discrete level by only О (N log k N) quantities. Thus, a matrix vector multiplication as a basic step in every iterative solver can be performed by О(N log k N) operations. This method can be applied to all kinds of integral equations discretized by, e.g., the Nystrom, the collocation or the Galerkin method.

In this paper we present cubature methods for the approximation of surface integrals arising from Galerkin discretizations of 3-d boundary integral equations. This numerical integrator is fully implicit in the sense that the form of the kernel function, the surface parametrization, the trial and test space, and the order of the singularity of the kernel function is not used explicitly. Different kernels can be treated by just replacing the subroutine which evalutes the kernel function in certain surface points.

The present work is devoted to the a posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. Only two global constants appear in the definition of the estimator; both constants depend solely on the domain geometry, and the estimator is quite nonsensitive to the error in the constants evaluation. It is also shown how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

The ILU iteration scheme is well known as an excellent smoother in a multigrid process. But up to now a restricting fact of the method was that, apparently, the algorithm can only be applied efficiently to finite-difference discretizations on rectangular grids. The problem to transfer the algorithm to finite-element discretizations is that the iteration depends on the numbering of the grid points and on the structure of the grid. In opposition to this, the basic advantage of finite elements is that one can use self-adaptive refinement strategies, to get problem-orientated grids, which have not a uniform structure. In this paper we explain how to apply the ILU method to arbitrary finite-element grids and develop strategies for accelerating the algorithm and making it vectorizable. In Section 3 we shall study the influence of the grid for the stability of the ILU iteration and give a somewhat surprising example, which makes us optimistic with regard to a generalization of the theoretical results to larger classes of problems. Finally, in Section 4 we report on some numerical tests for an eigenvalue problem with real physical background.