Matthias Maischak

Adaptive multilevel BEM for acoustic scattering

We study the h- and p-versions of the Galerkin boundary element method for integral equations of the first kind in 2D and 3D which result from the scattering of time harmonic acoustic waves at hard or soft scatterers. We derive an abstract a-posteriori error estimate for indefinite problems which is based on stable multilevel decompositions of our test and trial spaces. The Galerkin error is estimated by easily computable local error indicators and an adaptive algorithm for h- or p-adaptivity is formulated. The theoretical results are illustrated by numerical examples for hard and soft scatterers in 2D and 3D.

A posteriori error estimate and h-adaptive algorithm on surfaces for Symm's integral equation

Summary. A residual-based a posteriori error estimate for boundary integral equations on surfaces is derived in this paper. A localisation argument involves a Lipschitz partition of unity such as nodal basis functions known from finite element methods. The abstract estimate does not use any property of the discrete solution, but simplifies for the Galerkin discretisation of Symms integral equation if piecewise constants belong to the test space. The estimate suggests an isotropic adaptive algorithm for automatic mesh-refinement. An alternative motivation from a two-level error estimate is possible but then requires a saturation assumption. The efficiency of an anisotropic version is discussed and supported by numerical experiments.

Residual-based a posteriori error estimate for hypersingular equation on surfaces

Summary.The hypersingular integral equation of the first kind equivalently describes screen and Neumann problems on an open surface piece. The paper establishes a computable upper error bound for its Galerkin approximation and so motivates adaptive mesh refining algorithms. Numerical experiments for triangular elements on a screen provide empirical evidence of the superiority of adapted over uniform mesh-refining. The numerical realisation requires the evaluation of the hypersingular integral operator at a source point; this and other details on the algorithm are included.

Exponential convergence of the hp-version for the boundary element method on open surfaces

Summary. We analyze the boundary element Galerkin method for weakly singular and hypersingular integral equations of the first kind on open surfaces. We show that the hp-version of the Galerkin method with geometrically refined meshes converges exponentially fast for both integral equations. The proof of this fast convergence is based on the special structure of the solutions of the integral equations which possess specific singularities at the corners and the edges of the surface. We show that these singularities can be efficiently approximated by piecewise tensor products of splines of different degrees on geometrically graded meshes. Numerical experiments supporting these results are presented.

The hp-version of the boundary element method of Helmholtz screen problems

We study the boundary element method for weakly singular and hypersingular integral equations of the first kind on screens resulting from the Dirichlet and Neumann problems for the Helmholtz equation. It is shown that the hp-version with geometrical refined meshes converges exponentially fast in both cases. We underline our theoretical results by numerical experiments for the pure h-, p-versions, the graded mesh and the hp-version with geometrically refined mesh.ZusammenfassungWir betrachten die Randelementmethode für schwachsinguläre und hypersinguläre Integralgleichungen erster Art auf Schirmen. Die Integralgleichungen sind äquivalent zum Dirichlet- beziehungsweise Neumann-Problem für die Helmholtz-Gleichung im Außengebiet. Es wird gezeigt, daß die hp-Version mit geometrischem Gitter in beiden Fällen exponentiell in Abhängigkeit von den Freiheitsgraden konvergiert. Wir bestätigen unsere theoretischen Ergebnisse durch numerische Experimente für die reine h- und p-Version, für das graduierte Gitter und für die hp-Version mit geometrischem Gitter.

Multiplicative Schwarz Algorithms for the Galerkin Boundary Element Method

We study the multiplicative Schwarz method for the h- and the p-version Galerkin boundary element method for a hypersingular and a weakly singular integral equation of the first kind. For both integral equations we prove that the contraction rate of the multiplicative Schwarz operator is strictly less than 1 for the h-version for the two level and the multilevel methods, whereas for the p-version we show that the contraction rate approaches one only logarithmically in p for the 2-level method. Computational results are presented for both the h-version and the p-version which support our theory.

Preconditioned minimum residual iteration for the h–p version of the coupled FEM/BEM with quasi‐uniform meshes

We propose and analyze efficient preconditioners for the minimum residual method to solve indefinite, symmetric systems of equations arising from the h–p version of finite element and boundary element coupling. According to the structure of the Galerkin matrix we study two- and three-block preconditioners corresponding to Neumann and Dirichlet problems for the finite element discretization. In the case of exact inversion of the blocks we obtain bounded iteration numbers for the two-block Jacobi solver and O(h−3/4p3/2) iteration numbers for the three-block Jacobi solver. Here, h denotes the mesh size and p the polynomial degree. For the efficient two-block method we analyze the influence of various preconditioners which are based on further decomposing the trial functions into nodal, edge and interior functions. By further splitting the ansatz space with respect to basis functions associated with the edges we obtain a partially diagonal preconditioner. The penultimate method requires O(log2p) iterations whereas the latter method needs O(p log2p) iterations. Numerical results are presented which support the theory. Copyright

Adaptive time-domain boundary element methods and engineering applications

Parts of this work were funded by BMWi under the project SPERoN 2020, part II, Leiser StraB enverkehr, grant number 19 U 10016 F. H. G. acknowledges support by ERC Advanced Grant HARG 268105.

Adaptive FE–BE coupling for strongly nonlinear transmission problems with Coulomb friction

We analyze an adaptive finite element/boundary element procedure for scalar elastoplastic interface problems involving friction, where a nonlinear uniformly monotone operator such as the p-Laplacian is coupled to the linear Laplace equation on the exterior domain. The problem is reduced to a boundary/domain variational inequality, a discretized saddle point formulation of which is then solved using the Uzawa algorithm and adaptive mesh refinements based on a gradient recovery scheme. The Galerkin approximations are shown to converge to the unique solution of the variational problem in a suitable product of Lp- and L2-Sobolev spaces.

Finite element/boundary element coupling for two-body elastoplastic contact problems with friction

We consider two-body contact problems in elastoplasticity (plasticity with isotropic hardening) with and without friction and present solution procedures based on the coupling of finite elements and boundary elements. One solution method consists in rewriting the problem with penalty terms taking care of the frictional contact conditions [4], see also [8]. Then, its discretized version is solved by applying the radial return algorithm for both friction and plastification. We perform a segment-to-segment contact discretization which allows also to treat friction. Another solution procedure uses mortar projections [2] together with a Dirichlet-toNeumann (DtN) algorithm for the frictional contact part [6]; here we still use radial return for the plasticity part. Furthermore, extending the approach in [7] we can rewrite the contact problems with friction (given as variational inequalities without regularization) as saddle point problems and directly apply Uzawa’s algorithm. Comments are given for adaptive procedures [5]. Numerical benchmarks are given for small deformations and demonstrate the wide applicability of the given methods.