TY - JOUR
T1 - The $hp$-Version of BEM — Fast Convergence, Adaptivity and Efficient Preconditioning
AU - Ernst P. Stephan
JO - Journal of Computational Mathematics
VL - 2-3
SP - 348
EP - 359
PY - 2009
DA - 2009/04
SN - 27
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/10370.html
KW - $hp$-version of the boundary element method, Adaptive refinement, Preconditioning, Signorini contact.
AB - <p style="text-align: justify;">In this survey paper we report on recent developments of the $hp$-version of the boundary element method (BEM). As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar, open surface. We show that the Galerkin solutions (computed with the $hp$-version on geometric meshes) converge exponentially fast towards the exact solutions of the integral equations. An $hp$-adaptive algorithm is given and the implementation of the $hp$-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices. We also comment on the use of the $hp$-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points. Numerical results are presented which underline the theoretical results.</p>