Boundary-Element-Based Finite Element Methods for Helmholtz and Maxwell Equations on General Polyhedral Meshes
Authors: Dylan M. Copeland
We present new finite element methods for Helmholtz and Maxwell equations on general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1076764Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1421
 R.A. Adams. Sobolev spaces, volume 65 of Pure and Applied Mathematics. Academic Press
[a subsidiary of Harcourt Brace Jovanovich Publishers], New York-London, 1975.
 M. Bebendorf. Approximation of boundary element matrices. Numerische Mathematik, 86, pp. 565-589, 2000.
 F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM Journal on Numerical Analysis, 43, pp. 1872-1896 (electronic), 2005.
 A. Buffa, and S.H. Christiansen. The electric field integral equation on Lipschitz screens: definitions and numerical approximation. Numerische Mathematik, 94, pp. 229-267, 2003.
 A. Buffa and P. Ciarlet, Jr. On traces for functional spaces related to Maxwell-s equations. I. An integration by parts formula in Lipschitz polyhedra. Mathematical Methods in the Applied Sciences, 24, pp. 9-30, 2001.
 A. Buffa and P. Ciarlet, Jr. On traces for functional spaces related to Maxwell-s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Mathematical Methods in the Applied Sciences, 24, pp. 31-48, 2001.
 A. Buffa, M. Costabel, and C. Schwab. Boundary element methods for Maxwell-s equations on non-smooth domains. Numerische Mathematik, 92, pp. 679-710, 2002.
 A. Buffa, M. Costabel, and D. Sheen. On traces for H(curl, ╬®) in Lipschitz domains. Journal of Mathematical Analysis and Applications, 276, pp. 845-867, 2002.
 A. Buffa and R. Hiptmair. Galerkin boundary element methods for electromagnetic scattering. Topics in computational wave propagation, volume 31 of Lecture Notes in Computational Science and Engineering, pp. 83-124. Springer, Berlin, 2003.
 A. Buffa and R. Hiptmair. A coercive combined field integral equation for electromagnetic scattering. SIAM Journal on Numerical Analysis, 42, pp. 621-640 (electronic), 2004.
 A. Buffa and R. Hiptmair. Regularized combined field integral equations. Numerische Mathematik, 100, pp. 1-19, 2005.
 A. Buffa, R. Hiptmair, T. von Petersdorff, and C. Schwab. Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numerische Mathematik, 95, pp. 459-485, 2003.
 M. Costabel. Boundary integral operators on Lipschitz domains: elementary results. SIAM Journal on Mathematical Analysis, 19, pp. 613-626, 1988.
 V. Dolean, H. Fol, S. Lanteri, and R. Perrussel. Solution of the timeharmonic Maxwell equations using discontinuous Galerkin methods. Journal of Computational and Applied Mathematics, 218, pp. 435-445, 2008.
 S. Erichsen and S.A. Sauter. Efficient automatic quadrature in 3-d Galerkin BEM. Computer Methods in Applied Mechanics and Engineering, 157, pp. 215-224, 1998. Seventh Conference on Numerical Methods and Computational Mechanics in Science and Engineering (NMCM 96) (Miskolc).
 W. Hackbusch A sparse matrix arithmetic based on H-matrices. I. Introduction to H-matrices. Computing. Archives for Scientific Computing, 62, pp. 89-108, 1999.
 R. Hiptmair. Coupling of finite elements and boundary elements in electromagnetic scattering. SIAM Journal on Numerical Analysis, 41, pp. 919-944 (electronic), 2003.
 R. Hiptmair and P. Meury. Stabilized FEM-BEM coupling for Helmholtz transmission problems. SIAM Journal on Numerical Analysis, 44, pp. 2107-2130 (electronic), 2006.
 P. Houston, I. Perugia, A. Schneebeli, and D. Sch┬¿otzau. Interior penalty method for the indefinite time-harmonic Maxwell equations. Numerische Mathematik, 100, pp. 485-518, 2005.
 P. Houston, I. Perugia, A. Schneebeli, and D. Sch┬¿otzau. Mixed discontinuous Galerkin approximation of the Maxwell operator: the indefinite case. M2AN. Mathematical Modelling and Numerical Analysis, 39, pp. 727-753, 2005.
 G.C. Hsiao, O. Steinbach, and W.L. Wendland. Domain decomposition methods via boundary integral equations. Journal of Computational and Applied Mathematics, 125, pp. 521-537, 2000. Numerical analysis 2000, Vol. VI, Ordinary differential equations and integral equations.
 J.M. Hyman and M. Shashkov. Mimetic discretizations for Maxwell-s equations. Journal of Computational Physics, 151, pp. 881-909, 1999.
 Y. Kuznetsov, K. Lipnikov, and M .Shashkov. The mimetic finite difference method on polygonal meshes for diffusion-type problems. Computational Geosciences, 8, pp. 301-324, 2005.
 K. Lipnikov, M. Shashkov, and D. Svyatskiy. The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. Journal of Computational Physics, 211, pp. 473-491, 2006.
 W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge, 2000.
 P. Monk. Finite element methods for Maxwell-s equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.
 O. Steinbach. Numerical approximation methods for elliptic boundary value problems. Finite and boundary elements. Springer, New York, 2008. Translated from the 2003 German original.