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Volume 27, Issue 2-3
The $hp$-Version of BEM — Fast Convergence, Adaptivity and Efficient Preconditioning

Ernst P. Stephan

J. Comp. Math., 27 (2009), pp. 348-359.

Published online: 2009-04

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  • Abstract

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.

  • Keywords

$hp$-version of the boundary element method, Adaptive refinement, Preconditioning, Signorini contact.

  • AMS Subject Headings

65N55.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-348, author = {Ernst P. Stephan}, title = {The $hp$-Version of BEM — Fast Convergence, Adaptivity and Efficient Preconditioning}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {348--359}, abstract = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10370.html} }
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 -

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.

Ernst P. Stephan. (2009). The $hp$-Version of BEM — Fast Convergence, Adaptivity and Efficient Preconditioning. Journal of Computational Mathematics. 27 (2-3). 348-359. doi:
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