Band 23, Heft 4 - Special Issue Dedicated to Professor Thomas Apel on the Occasion of his 60th Birthday, Guest Editors: Sven Beuchler and Arnd Rösch

We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.

Recently, H ⁢ ( div ) \mathbf{H}(\mathrm{div}) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case. In the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart–Thomas elements and higher-order Lagrange-elements. This ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.

In this work, a cost-efficient space-time adaptive algorithm based on the Dual Weighted Residual (DWR) method is developed and studied for a coupled model problem of flow and convection-dominated transport. Key ingredients are a multirate approach adapted to varying dynamics in time of the subproblems, weighted and non-weighted error indicators for the transport and flow problem, respectively, and the concept of space-time slabs based on tensor product spaces for the data structure. In numerical examples, the performance of the underlying algorithm is studied for benchmark problems and applications of practical interest. Moreover, the interaction of stabilization and goal-oriented adaptivity is investigated for strongly convection-dominated transport.

We study the numerical approximation of a control problem governed by a semilinear parabolic problem, where the usual Tikhonov regularization term in the cost functional is replaced by a non-differentiable sparsity-promoting term.

We consider Helmholtz problems in three-dimensional domains featuring conical points. We focus on the high-frequency regime and derive novel sharp upper-bounds for the stress intensity factors of the singularities associated with the conical points. We then employ these new estimates to analyse the stability of finite element discretisations. Our key result is that lowest-order Lagrange finite elements are stable under the assumption that “ ω 2 ⁢ h \omega^{2}h is small”. This assumption is standard and well known in the case of smooth domains, and we show that it naturally extends to domain with conical points, even when using uniform meshes.

In this paper we discuss the use of implicit Runge–Kutta schemes for the time discretization of optimal control problems with evolution equations. The specialty of the considered discretizations is that the discretizations schemes for the state and adjoint state are chosen such that discretization and optimization commute. It is well known that for Runge–Kutta schemes with this property additional order conditions are necessary. We give sufficient conditions for which class of schemes these additional order condition are automatically fulfilled. The focus is especially on implicit Runge–Kutta schemes of Gauss, Radau IA, Radau IIA, Lobatto IIIA, Lobatto IIIB and Lobatto IIIC collocation type up to order six. Furthermore, we also use a SDIRK (singly diagonally implicit Runge–Kutta) method to demonstrate, that for general implicit Runge–Kutta methods the additional order conditions are not automatically fulfilled. Numerical examples illustrate the predicted convergence rates.

Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

Numerical studies are presented that investigate finite element methods satisfying discrete maximum principles for convection-diffusion-reaction equations. Two linear methods and several nonlinear schemes, some of them proposed only recently, are included in these studies, which consider a number of two-dimensional examples. The evaluation of the results examines the accuracy of the numerical solutions with respect to quantities of interest, like layer widths, and the efficiency of the simulations.

We consider standard tracking-type, distributed elliptic optimal control problems with L 2 L^{2} regularization, and their finite element discretization. We are investigating the L 2 L^{2} error between the finite element approximation u ϱ ⁢ h u_{\varrho h} of the state u ϱ u_{\varrho} and the desired state (target) u ¯ \overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ = h 4 \varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.

This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ ( Ω ) {H^{2}(\Omega)} but only to H r ⁢ ( Ω ) {H^{r}(\Omega)} for some r ∈ ( 1 , 2 ) {r\in(1,2)} . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} and H 1 ⁢ ( Ω ) {H^{1}(\Omega)} . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results.