The hp-version of the boundary element method for 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.     

Schnelle Drehmomentregelung im gesamten Drehzahlbereich eines hochausgenutzten Drehfeldantriebs

Electrical Engineering - Das jeweilige maximale Drehmoment einer wechselrichtergespeisten Kurzschlußläufer-Asynchronmaschine ist durch die Begrenzung von Wechselrichter-Ausgangsstrom und...     

FE/BE coupling for an acoustic fluid–structure interaction problem. Residual a posteriori error estimates

In this paper, we developed an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid–structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time‐harmonic vibration. Our methods combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well‐known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two‐dimensional case we performed an adaptive algorithm on triangles, and for the three‐dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

An <Emphasis Type="Italic">hp</Emphasis>-adaptive finite element/boundary element coupling method for electromagnetic problems

We present an hp-version of the finite element / boundary element coupling method to solve the eddy current problem for the time-harmonic Maxwell’s equations. We use H(curl, Ω -conforming vector-valued polynomials to approximate the electric field in the conductor Ω and surface curls of continuous piecewise polynomials on the boundary Γ of Ω to approximate the twisted tangential trace of the magnetic field on Γ. We present both a priori and a posteriori error estimates together with a three-fold hp-adaptive algorithm to compute the fem/bem coupling solution with appropriate distributions of polynomial degrees on suitably refined meshes.     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

Some schwarz methods for integral equations on surfaces-h and p versions

Abstact We present new results from 11, 7, 12 on various Schwarz methods for the h and p versions of the boundary element methods applied to prototype first kind integral equations on surfaces. When those integral equations (weakly/hypersingular) are solved numerically by the Galerkin boundary element method, the resulting matrices become ill-conditioned. Hence, for an efficient solution procedure appropriate preconditioners are necessary to reduce the numbers of CG-iterations. In the p version where accuracy of the Galerkin solution is achieved by increasing the polynomial degree the use of suitable Schwarz preconditioners (presented in the paper) leads to only polylogarithmically growing condition numbers. For the h version where accuracy is achieved by reducing the mesh size we present a multi-level additive Schwarz method which is competitive with the multigrid method. Communicated by: U. Langer Dedicated to George C. Hsiao on the occasion of his 70th birthday.     

High‐order space‐time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space‐time coupling matrices are diagonalizable over for r ?100, and this means that the time‐coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG‐in‐time methodology, for the first time, to second‐order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high‐order (up to degree 7) temporal and spatio‐temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.