Past Event: Oden Institute Seminar

Advances in High Order Boundary Element Methods

Lucy Weggler, Saarland University, Germany

Abstract

The use of high order approximation methods is very effective in achieving high accuracy numerical simulations while keeping the number of unknowns moderate, in articular for piece- wise smooth solutions of partial differential equations. The numerical and theoretical studies for this kind of methods began in the late 70s - early 80s. Since that time high order dis- cretization schemes are getting increasingly popular in many practical applications such as fluid dynamics, structural mechanics, electromagnetics, acoustics, etc. Nowadays, one can say that the high order methods are an established field of research in the finite element community [1, 2]. In what concerns the boundary element community, most recent publications show that theory and numerics of the high order boundary element methods are getting more and more interesting [3, 4, 5, 6]. This talk is concerned with the development and application of high order boundary elements as presented in [7]. After shortly recalling the theoretical results on the high order convergence rates for Galerkin solutions, the key ideas behind the high order boundary element implementa- tion are discussed. At the one hand, this is the abstract relation between energy spaces and trace spaces that appear in variational formulations of elliptic and Maxwell boundary value problems. On the other hand, this is a general access to the definition of curved element shapes that go along well with the high order basis functions needed to discretize the variational formulations resulting from a boundary integral equation. In the second part of this talk we consider the problem of electromagnetic scattering at the perfect electric conductor. Numerical results for the high order boundary element methods are presented. Our tests bring awareness of the necessity to enable a high order description of the geometry. This, in turn, gives rise to ongoing work on theoretical and practical tasks that come into play here, i.e., the approximation theory of finite-dimensional spaces of tangential vector fields on curved manifolds or isogeometric analysis in general. References [1] Demkowicz, L., Computing with hp-adaptive finite elements. Vol. 1, Chapman & Hall/CRC Applied Math. and Nonlin. Science Series, 2007. [2] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz and A. Zdunek, Computing with hp-adaptive finite elements. Vol. 2, Chapmen & Hall/CRC Applied Math. and Nonlin. Science Series, 2008. [3] E.P. Stephan, M. Maischak and F. Leydecker, An hp-adaptive fem/bem coupling method for electromagnetic problems, Comput. Mech, Vol. 39, 2007. [4] M. Maischak and E.P. Stephan, Adaptive hp-version of boundary element methods for elastic contact problems, Comput. Mech, Vol. 39, 2007. [5] S. Sauter and Ch. Schwab, Boundary element methods, Vol. 39, Comp. Math., Springer, 2011 (translated from the German original published in 2004). [6] A. Bespalov, N. Heuer and R. Hiptmair, Convergence of the Natural hp-BEM for the Electric Field Integral Equation on Polyhedral Surfaces., J. Numer. Anal., Vol 48, 2010. [7] L. Weggler, High Order Boundary Element Methods, Dissertation, Saarland University, 2011. Bio: Lucy Weggler, born in 1981 in Muhlacker, Germany, started studying mechanical engineering and mathematics at Saarland University in 2000. Having obtained a DAAD scholarship for study abroad, she completed a one year stay at the University of Texas at Austin in 2008. The academic mentoring was offered by Prof. Dr. Leszek Demkowicz. Since then, Lucy Weggler's mathematical research concentrated on the development and application of high order boundary element methods. The doctoral thesis supervised by Prof. Dr. Sergej Rjasanow, was completed in December 2011 and awarded the highest rating "summa com laude".