Special Issue

Applied Numerical Mathematics 59 (2009) 2695–2697

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

Preface

Special Issue Boundary Elements – Theory and Applications Dedicated to Professor Ernst P. Stephan on the Occasion of his 60th Birthday

This special issue is dedicated to Professor Ernst P. Stephan on the occasion of his 60th birthday. It contains selected papers from the conference Boundary Elements – Theory and Applications (BETA) that took place in Hannover, Germany, 22–24 May 2007, and was held to celebrate Prof. Stephan’s birthday. One dominant feature of the BETA conference was the presence of E.P. Stephan’s many past and present co-authors which reflects his extreme productivity and success in establishing research contacts with people from around the world. E.P. Stephan is a leading mathematician in applied mathematics and has the gift and expertise to combine research in modern mathematical analysis with mathematical physics and engineering to simulate and analyse problems of theses areas. E.P. Stephan was born in Birkenau, a small village in the state of Hesse in south-west Germany (18.5.1947). In 1970 he obtained his diploma at the Technical University Darmstadt in Germany (Technische Hochschule Darmstadt, THD, at that time) and subsequently his doctoral degree (1975) and his habilitation (1984), both at THD under W.L. Wendland. In his pioneering doctoral thesis [17] he showed stability and discrete convergence of finite difference approximations for strongly elliptic pseudodifferential equations in Rn . In his paper with V. Weissgerber [19] he derived a new stability result on hybrid finite element approximations to shell problems. At that time he began to work on finite element methods for singularity problems [13]. Together with his supervisor, W.L. Wendland, and G.C. Hsiao, E.P. Stephan was influential in the development of the boundary element method (BEM) with their fundamental contribution [20] on the convergence of Galerkin and least squares approximations of pseudo-differential operators and the treatment of systems of integral equations stemming from mixed boundary conditions [23]. Later, the use of the Mellin transformation was established by M. Costabel and E.P. Stephan as a ground-breaking tool for the analysis of boundary integral operators on polygonal domains in the framework of Sobolev 0168-9274/$30.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2008.12.022

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Preface / Applied Numerical Mathematics 59 (2009) 2695–2697

spaces. The publication of the seminal paper with M. Costabel [4] was delayed for a few years due to the political circumstances in Poland. During his Darmstadt period, E.P. Stephan was invited by G.J. Fix to the Carnegie-Mellon University in Pittsburgh where they studied finite element least squares approximations of fourth-order problems [7]. There he also met R.C. MacCamy and together they developed an early analysis of boundary integral equations applied to electro-magnetic scattering problems [11]. In 1983, before having finished his habilitation thesis, E.P. Stephan became Associate Professor at the Georgia Institute of Technology, Atlanta, and was promoted to Full Professor in 1988. In this period, E.P. Stephan analysed boundary integral operators for crack and screen problems [14]. During a visit to the University of Maryland, and under the influence of I. Babuška, he and M. Suri implemented the use of the p- and hp-versions for the BEM, see e.g. [18]. These were the very first steps in establishing high-order boundary element Galerkin methods for the solution of integral equations of the first kind in which E.P. Stephan was and is the dominating figure [15]. With his student T. von Petersdorff, who won the von-Mises prize of the GAMM for his PhD thesis in 1991, E.P. Stephan analysed three-dimensional edge and corner singularities and their approximation on graded meshes [22]. Besides regularity and approximation properties, they also looked at the challenging problem of the efficient iterative solution of the highdimensional discrete systems stemming from the BEM and established first multigrid methods for integral equations of the first kind in [21]. During this period, the symmetric coupling of the finite and boundary element method was invented by M. Costabel and jointly developed with E.P. Stephan as an efficient procedure for the solution of a wide class of boundary value problems [5,6]. In [16], E.P. Stephan gives an overview of this area, including the contributions of his group. In 1989, E.P. Stephan accepted the offer of a full professorship at the University of Hannover (now Leibniz Universität Hannover) where he continues until today. Motivated by the residual-based adaptive finite element method and its emerging mathematical theory, E.P. Stephan and his Hannover research group established the first explicit residual-based a posteriori error estimate for integral equations with norms in fractional-order Sobolev spaces, from simple model examples [1] to hypersingular operators on screens [2]. In the same period they developed a regularity and approximation theory for hp-methods applied to boundary integral equation on polygons [8]. This work culminated in a proof of the exponential convergence of the hp-version of the BEM on open surfaces [10]. With the surge of domain decomposition methods in the late 90s, E.P. Stephan implemented their use for preconditioned iterative solvers and developed and analysed Schwarz-type methods for the BEM [9,12]. A recent development is the analysis and numerical treatment of contact problems which give rise to variational inequalities. In [3], E.P. Stephan and co-authors demonstrate the efficiency of the BEM for such problems and establish a framework where hp-methods show their potential. E.P. Stephan is a continuous source of new ideas and has influenced many researchers, in particular young people. He has guided 17 doctoral students and three people of his group, the co-editors of this special issue, received their habilitations under his supervision. E.P. Stephan is our good friend and of many of our colleagues. We are grateful for his guidance and collegiality throughout the years. We wish him many productive years to come and a long and happy future with his wife Karin. October 2008 Guest Editors Carsten Carstensen Humboldt-Universität zu Berlin, Germany Norbert Heuer Pontificia Universidad Católica de Chile, Santiago, Chile Matthias Maischak Brunel University, Uxbridge, United Kingdom Available online 25 December 2008

References [1] C. Carstensen, E.P. Stephan, A posteriori error estimates for boundary element methods, Mathematics of Computation 64 (1995) 483–500. [2] C. Carstensen, M. Maischak, D. Praetorius, E.P. Stephan, Residual-based a posteriori error estimate for hypersingular equation on surfaces, Numerische Mathematik 97 (2004) 397–425. [3] A. Chernov, M. Maischak, E.P. Stephan, hp-mortar boundary element method for two-body contact problems with friction, Mathematical Methods in the Applied Sciences 31 (2008) 2029–2054. [4] M. Costabel, E.P. Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, in: Mathematical Models and Methods in Mechanics, in: Banach Center Publications, vol. 15, Polish. Acad. Sci., Warsaw, 1985, pp. 175–251. [5] M. Costabel, E.P. Stephan, Coupling of finite elements and boundary elements for inhomogeneous transmission problems in R3 , in: The Mathematics of Finite Elements and Applications, VI, Uxbridge, 1987, Academic Press, London, 1988, pp. 289–296. [6] M. Costabel, E.P. Stephan, Coupling of finite element and boundary element methods for an elasto-plastic interface problem, SIAM Journal on Numerical Analysis 27 (1990) 1212–1226. [7] G.J. Fix, E.P. Stephan, On the finite element-least squares approximation to higher order elliptic systems, Archive for Rational Mechanics and Analysis 91 (1985) 137–151. [8] N. Heuer, E.P. Stephan, Boundary integral operators in countably normed spaces, Mathematische Nachrichten 191 (1998) 123–151.

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[9] N. Heuer, E.P. Stephan, An additive Schwarz method for the h-p version of the boundary element method for hypersingular integral equations in R3 , IMA Journal of Numerical Analysis 21 (2001) 265–283. [10] N. Heuer, M. Maischak, E.P. Stephan, Exponential convergence of the hp-version for the boundary element method on open surfaces, Numerische Mathematik 83 (1999) 641–666. [11] R.C. MacCamy, E.P. Stephan, A boundary element method for an exterior problem for three-dimensional Maxwell’s equations, Applicable Analysis 16 (1983) 141–163. [12] M. Maischak, E.P. Stephan, T. Tran, Multiplicative Schwarz algorithms for the Galerkin boundary element method, SIAM Journal on Numerical Analysis 38 (2000) 1243–1268. [13] E. Stephan, Conform and mixed finite element schemes for the Dirichlet problem for the bi-Laplacian in plane domains with corners, Mathematical Methods in the Applied Sciences 1 (1979) 354–382. [14] E.P. Stephan, Boundary integral equations for screen problems, Journal of Integral Equations and Operator Theory 10 (1987) 236–257. [15] E.P. Stephan, The h-p version of the boundary element method for solving 2- and 3-dimensional problems, Computer Methods in Applied Mechanics and Engineering 133 (1996) 183–208. [16] E.P. Stephan, Coupling of boundary element methods and finite element methods, in: E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Fundamentals, in: Encyclopedia of Computational Mechanics, vol. 1, John Wiley & Sons, 2004 (Chapter 13). [17] E.P. Stephan, Differenzenapproximationen von Pseudodifferentialoperatoren, PhD thesis, Technische Hochschule Darmstadt, 1975. [18] E.P. Stephan, M. Suri, On the convergence of the p-version for some boundary element Galerkin methods, Mathematics of Computation 52 (1989) 31–48. [19] E.P. Stephan, V. Weissgerber, Zur Approximation von Schalen mit hybriden Elementen, Computing 20 (1978) 75–94. [20] E.P. Stephan, W.L. Wendland, Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, Manuscripta Geodaetica 1 (1976) 93–123. [21] T. von Petersdorff, E.P. Stephan, On the convergence of the multigrid method for a hyper-singular integral equation of the first kind, Numerische Mathematik 57 (1990) 379–391. [22] T. von Petersdorff, E.P. Stephan, Regularity of mixed boundary value problems in R3 and boundary element methods on graded meshes, Mathematical Methods in the Applied Sciences 12 (1990) 229–249. [23] W.L. Wendland, E.P. Stephan, G.C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Mathematical Methods in the Applied Sciences 1 (1979) 265–321.