Band 14, Heft 4

In this paper we consider the transmission eigenvalue problem corresponding to acoustic scattering by a bounded isotropic inhomogeneous object in two dimensions. This is a non-self-adjoint eigenvalue problem for a quadratic pencil of operators. In particular we are concerned with theoretical error analysis of a finite element method for computing the eigenvalues and corresponding eigenfunctions. Our analysis of convergence makes use of Osborn's perturbation theory for eigenvalues of non-self-adjoint compact operators. Some numerical examples are presented to confirm our theoretical error analysis.

In this paper, an a posteriori residual error estimator is proposed for the A−ϕ magnetodynamic Maxwell system given in its potential and space/time formulation and solved by a finite element method. The reliability as well as the efficiency of the estimator are established for several norms. Then, numerical tests are performed, allowing to illustrate the obtained theoretical results.

We develop an arbitrary-order primal method for diffusion problems on general polyhedral meshes. The degrees of freedom are scalar-valued polynomials of the same order at mesh elements and faces. The cornerstone of the method is a local (elementwise) discrete gradient reconstruction operator. The design of the method additionally hinges on a least-squares penalty term on faces weakly enforcing the matching between local element- and face-based degrees of freedom. The scheme is proved to optimally converge in the energy norm and in the L 2 -norm of the potential for smooth solutions. In the lowest-order case, equivalence with the Hybrid Finite Volume method is shown. The theoretical results are confirmed by numerical experiments up to order 4 on several polygonal meshes.

We consider a finite volume method for flow simulations in an anisotropic porous medium in the presence of a well. The hydraulic head varies logarithmically and its gradient changes rapidly in the well vicinity. Thus, the use of standard numerical schemes results in a completely wrong total well flux and an inaccurate hydraulic head. In this article we propose two finite volume methods to model the well singularity in an anisotropic medium. The first method significantly reduces the total well flux error, but the hydraulic head is still not even first-order accurate. The second method gives a second-order accurate hydraulic head and at least first-order accurate total well flux.

We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.

We propose a superfast discrete Haar wavelet transform (SFHWT) as well as its inverse, using the low-rank Quantics-TT (QTT) representation for the Haar transform matrices and input-output vectors. Though the Haar matrix itself does not have a low QTT rank approximation, we show that factor matrices used at each step of the traditional multilevel Haar wavelet transform algorithm have explicit QTT representations of low rank. The SFHWT applies to a vector representing a signal sampled on a uniform grid of size N=2d${N=2^d}$. We develop two algorithms which roughly require square logarithmic time complexity O(log2N)${O(\log ^2 N)}$ with respect to the grid size, hence outperforming the traditional fast Haar wavelet transform (FHWT) of linear complexity O ( N ). Our approach also applies to the FHWT inverse as well as to the multidimensional wavelet transform. Numerical experiments demonstrate that the SFHWT algorithm is robust in keeping low rank of the resulting output vector and it outperforms the traditional FHWT for grid sizes larger than a certain value depending on the spacial dimension.

Two optimal control problems for instationary magnetization processes are considered in 3D spatial domains that include electrically conducting and nonconducting regions. The magnetic fields are generated by induction coils. In the first model, the induction coil is considered as part of the conducting region and the electrical current is taken as control. In the second, the coil is viewed as part of the nonconducting region and the electrical voltage is the control. Here, an integro-differential equation accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil. We derive first-order necessary optimality conditions for the optimal controls of both problems. Based on them, numerical methods of gradient type are applied. Moreover, we report on the application of model reduction by POD that lead to tremendous savings. Numerical tests are presented for academic 3D geometries but also for a real-world application.