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In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE-MG)

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 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.

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.