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A posteriori error estimates for the electric field integral equation on polyhedra. (English) Zbl 1416.65309

Chetverushkin, B. N. (ed.) et al., Contributions to partial differential equations and applications. Invited papers of the conferences ‘Contributions to partial differential equations’, Université Pierre et Marie Curie, Paris, France, August 31 – September 1, 2015 and ‘Applied and computational mathematics’, University of Houston, Texas, USA, February 26–27, 2016. Cham: Springer. Comput. Methods Appl. Sci. 47, 371-394 (2019).
Summary: We present a residual-based a posteriori error estimate for the Electric Field Integral Equation (EFIE) on a bounded polyhedron \(\varOmega \) with boundary \(\varGamma \). The EFIE is a variational equation formulated in \(\boldsymbol{H}^{-1/2}_{\mathrm{div}}(\varGamma)\). We express the estimate in terms of \(L^2\)-computable quantities and derive global lower and upper bounds (up to oscillation terms).
For the entire collection see [Zbl 1411.35011].

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65R20 Numerical methods for integral equations
78M25 Numerical methods in optics (MSC2010)
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[1] Alonso A (1996) Error estimators for a mixed method. Numer Math 74(4):385-395 · Zbl 0866.65068 · doi:10.1007/s002110050222
[2] Aurada M, Ferraz-Leite S, Praetorius D (2012) Estimator reduction and convergence of adaptive BEM. Appl Numer Math 62(6):787-801 · Zbl 1237.65131 · doi:10.1016/j.apnum.2011.06.014
[3] Beck R, Hiptmair R, Hoppe R, Wohlmuth B (2000) Residual based a posteriori error estimators for eddy current computation. M2AN Math Model Numer Anal 34(1):159-182 · Zbl 0949.65113 · doi:10.1051/m2an:2000136
[4] Bernardi C, Hecht F (2007) Quelques propriétés d’approximation des éléments finis de Nédélec, application à l’analyse a posteriori. C R Math Acad Sci Paris 344(7):461-466 · Zbl 1114.65139 · doi:10.1016/j.crma.2007.02.010
[5] Brakhage H, Werner P (1965) Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch Math 16:325-329 · Zbl 0132.33601 · doi:10.1007/BF01220037
[6] Buffa A, Ciarlet P Jr (2001) On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra. Math Methods Appl Sci 24(1):9-30 · Zbl 0998.46012 · doi:10.1002/1099-1476(20010110)24:1<9::AID-MMA191>3.0.CO;2-2
[7] Buffa A, Ciarlet P Jr (2001) On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math Methods Appl Sci 24(1):31-48 · Zbl 0976.46023 · doi:10.1002/1099-1476(20010110)24:1<31::AID-MMA193>3.0.CO;2-X
[8] Buffa A, Costabel M, Schwab C (2002) Boundary element methods for Maxwell’s equations on non-smooth domains. Numer Math 92(4):679-710 · Zbl 1019.65094 · doi:10.1007/s002110100372
[9] Buffa A, Costabel M, Sheen D (2002) On traces for \({H}(\text{ curl },\Omega )\) in Lipschitz domains. J Math Anal Appl 276(2):845-867 · Zbl 1106.35304
[10] Buffa A, Hiptmair R (2004) A coercive combined field integral equation for electromagnetic scattering. SIAM J Numer Anal 42(2):621-640 · Zbl 1082.78003 · doi:10.1137/S0036142903423393
[11] Buffa A, Hiptmair R (2005) Regularized combined field integral equations. Numer Math 100(1):1-19 · Zbl 1067.65137 · doi:10.1007/s00211-004-0579-9
[12] Buffa A, Hiptmair R, von Petersdorff T, Schwab C (2003) Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer Math 95(3):459-485 · Zbl 1071.65160 · doi:10.1007/s00211-002-0407-z
[13] Burton AJ, Miller GF (1971) The application of integral equation methods to the numerical solution of some exterior boundary-value problems. Proc Roy Soc London Ser A 323(1553):201-210 · Zbl 0235.65080 · doi:10.1098/rspa.1971.0097
[14] Carstensen C (1996) Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes. Math Comp 65(213):69-84 · Zbl 0849.65084 · doi:10.1090/S0025-5718-96-00671-0
[15] Carstensen C (1997) An a posteriori error estimate for a first-kind integral equation. Math Comp 66(217):139-155 · Zbl 0854.65102 · doi:10.1090/S0025-5718-97-00790-4
[16] Carstensen C, Faermann B (2001) Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind. Eng Anal Bound Elem 25(7):497-509 · Zbl 1007.65100 · doi:10.1016/S0955-7997(01)00012-1
[17] Carstensen C, Funken SA, Stephan EP (1996) A posteriori error estimates for \(hp\)-boundary element methods. Appl Anal 61(3-4):233-253 · Zbl 0882.65103
[18] Carstensen C, Maischak M, Praetorius D, Stephan EP (2004) Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer Math 97(3):397-425 · Zbl 1063.65116 · doi:10.1007/s00211-003-0506-5
[19] Carstensen C, Maischak M, Stephan EP (2001) A posteriori error estimate and \(h\)-adaptive algorithm on surfaces for Symm’s integral equation. Numer Math 90(2):197-213 · Zbl 1018.65138
[20] Carstensen C, Praetorius D (2012) Convergence of adaptive boundary element methods. J Integr Eqn Appl 24(1):1-23 · Zbl 1238.65124 · doi:10.1216/JIE-2012-24-1-1
[21] Carstensen C, Stephan EP (1995) A posteriori error estimates for boundary element methods. Math Comp 64(210):483-500 · Zbl 0831.65120 · doi:10.1090/S0025-5718-1995-1277764-7
[22] Cascon JM, Kreuzer C, Nochetto RH, Siebert KG (2008) Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer Anal 46(5):2524-2550 · Zbl 1176.65122 · doi:10.1137/07069047X
[23] Cascon JM, Nochetto RH, Siebert KG (2007) Design and convergence of AFEM in \(H(\rm{div})\). Math Models Methods Appl Sci 17(11):1849-1881 · Zbl 1144.65064
[24] Christiansen SH (2004) Discrete Fredholm properties and convergence estimates for the electric field integral equation. Math Comp 73(245):143-167 · Zbl 1034.65089 · doi:10.1090/S0025-5718-03-01581-3
[25] Clément P (1975) Approximation by finite element functions using local regularization. Rev Française Automat Informat Recherche Opérationnelle Sér RAIRO Anal Numér 9(R-2):77-84 · Zbl 0368.65008
[26] Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory, vol 93 of applied mathematical sciences, 2nd edn. Springer, Berlin · Zbl 0893.35138 · doi:10.1007/978-3-662-03537-5
[27] Demlow A, Dziuk G (2007) An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J Numer Anal 45(1):421-442 · Zbl 1160.65058 · doi:10.1137/050642873
[28] Erath C, Ferraz-Leite S, Funken S, Praetorius D (2009) Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl Numer Math 59(11):2713-2734 · Zbl 1177.65192 · doi:10.1016/j.apnum.2008.12.024
[29] Ern A, Guermond J-L (2004) Theory and practice of finite elements, vol 159. Applied mathematical sciences. Springer, New York · Zbl 1059.65103 · doi:10.1007/978-1-4757-4355-5
[30] Feischl M, Karkulik M, Melenk JM, Praetorius D (2013) Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J Numer Anal 51(2):1327-1348 · Zbl 1273.65186 · doi:10.1137/110842569
[31] Ferraz-Leite S, Ortner C, Praetorius D (2010) Convergence of simple adaptive Galerkin schemes based on \(h-h/2\) error estimators. Numer Math 116(2):291-316 · Zbl 1198.65213
[32] Ferraz-Leite S, Praetorius D (2008) Simple a posteriori error estimators for the \(h\)-version of the boundary element method. Computing 83(4):135-162 · Zbl 1175.65126
[33] Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73(2):325-348 · Zbl 0629.65005 · doi:10.1016/0021-9991(87)90140-9
[34] Greengard L, Rokhlin V (1997) A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numer 6:229-269 · Zbl 0889.65115 · doi:10.1017/S0962492900002725
[35] Hackbusch W, Nowak ZP (1989) On the fast matrix multiplication in the boundary element method by panel clustering. Numer Math 54(4):463-491 · Zbl 0641.65038 · doi:10.1007/BF01396324
[36] Hackbush W, Sauter SA (1993) On the efficient use of the Galerkin method to solve Fredholm integral equations. Appl Math 38(4-5):301-322 (Proceedings of ISNA ’92 - International Symposium on Numerical Analysis, Part I (Prague, 1992))
[37] Hiptmair R, Schwab C (2002) Natural boundary element methods for the electric field integral equation on polyhedra. SIAM J Numer Anal 40(1):66-86 · Zbl 1010.78014 · doi:10.1137/S0036142901387580
[38] Leis R (1965) Zur Dirichletschen Randwertaufgabe des Aussenraumes der Schwingungsgleichung. Math Z 90(3):205-211 · Zbl 0132.33602 · doi:10.1007/BF01119203
[39] Leydecker F, Maischak M, Stephan EP, Teltscher M (2010) Adaptive FE-BE coupling for an electromagnetic problem in \(\mathbb{R}^3\) - A residual error estimator. Math Methods Appl Sci 33(18):2162-2186 · Zbl 1227.78022
[40] Nochetto RH, Siebert KG, Veeser A (2009) Theory of adaptive finite element methods: an introduction. Multiscale. Nonlinear and Adaptive Approximation. Springer, Berlin, pp 409-542 · Zbl 1190.65176
[41] Nochetto RH, von Petersdorff T, Zhang C-S (2010) A posteriori error analysis for a class of integral equations and variational inequalities. Numer Math 116(3):519-552 · Zbl 1202.65085 · doi:10.1007/s00211-010-0310-y
[42] Nowak ZP, Hackbusch W (1986) Complexity of the method of panels. In: Marchuk G (ed) Computational processes and systems, No. 6. Nauka, Moscow, pp 233-244 (in Russian)
[43] Panich OI (1965) On the solubility of exterior boundary-value problems for the wave equation and for a system of Maxwell’s equations. Uspekhi Mat. Nauk 20:1(121):221-226
[44] Raviart P-A, Thomas JM (1977) A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods (Proceedings of Conference, Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), vol 606 of lecture notes in mathematics. Springer, Berlin, pp 292-315
[45] Rokhlin V (1985) Rapid solution of integral equations of classical potential theory. J Comput Phys 60(2):187-207 · Zbl 0629.65122 · doi:10.1016/0021-9991(85)90002-6
[46] Sauter SA, Schwab C (2011) Boundary element methods, vol 39. Springer series in computational mathematics. Springer, Berlin
[47] Tartar L (2007) An introduction to Sobolev spaces and interpolation spaces, vol 3. Lecture Notes of the Unione Matematica Italiana. Springer, Berlin · Zbl 1126.46001
[48] Teltscher M, Stephan EP, Maischak M (2003) A residual error estimator for an electromagnetic FEM-BEM coupling problem in \(\mathbb{R}^3\). Technical report, Institut für Angewandte Mathematik, Universität Hannover · Zbl 1227.78022
[49] Verfürth R (1998) A posteriori error estimators for convection-diffusion equations. Numer Math 80(4):641-663 · Zbl 0913.65095 · doi:10.1007/s002110050381
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