Volume 18, Issue 3

In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.

In this note we derive a posteriori error bounds for FE-discretisations for a fluid problem with cavitation. The underlying model is the Stokes system together with an inequality constraint for the pressure. In order to avoid suboptimal behavior of the error bounds we propose to employ a Lagrange setting yielding an improved estimate. Numerical tests confirm our theoretical results.

In this article we analyze a finite element method for solving H (div;Ω)-elliptic interface problems in general three-dimensional Lipschitz domains with smooth material interfaces. The continuous problems are discretized by means of lowest order H (div;Ω)-conforming finite elements of the first family (Raviart–Thomas or Nédélec face elements) on a family of unstructured oriented tetrahedral meshes. These resolve the smooth interface in the sense of sufficient approximation in terms of a parameter δ that quantifies the mismatch between the smooth interface and the finite element mesh. Optimal error estimates in the H (div;Ω)-norms are obtained for the first time. The analysis is based on a so-called δ -strip argument, a new extension theorem for H 1 (div)-functions across smooth interfaces, a novel non-standard interfaceaware interpolation operator, and a perturbation argument for degrees of freedom in H (div;Ω)-conforming finite elements. Numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution.

In the paper a balanced Neumann–Neumann algorithm for the reduced HCT finite element discretization on nonmatching meshes is discussed. The overall discretization is done using a mortar technique which is based on the application of an approximate matching condition for the discrete functions. The algorithms are analyzed using the abstract Schwarz framework, proving an almost optimal condition bound which is independent of the parameters of the problem, and depends only logarithmically on the ratio between the subdomain size and the mesh size.