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Adaptive boundary element methods for the computation of the electrostatic capacity on complex polyhedra. (English) Zbl 1453.78012

Summary: The accurate computation of the electrostatic capacity of three dimensional objects is a fascinating benchmark problem with a long and rich history. In particular, the capacity of the unit cube has widely been studied, and recent advances allow to compute its capacity to more than ten digits of accuracy. However, the accurate computation of the capacity for general three dimensional polyhedra is still an open problem. In this paper, we propose a new algorithm based on a combination of ZZ-type a posteriori error estimation and effective operator preconditioned boundary integral formulations to easily compute the capacity of complex three dimensional polyhedra to 5 digits and more. While this paper focuses on the capacity as a benchmark problem, it also discusses implementational issues of adaptive boundary element solvers, and we provide codes based on the boundary element package Bempp to make the underlying techniques accessible to a wide range of practical problems.

MSC:

78M15 Boundary element methods applied to problems in optics and electromagnetic theory
78A30 Electro- and magnetostatics
65N38 Boundary element methods for boundary value problems involving PDEs
65Z05 Applications to the sciences
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] Aurada, Markus; Feischl, Michael; Führer, Thomas; Karkulik, Michael; Melenk, J. Markus; Praetorius, Dirk, Local inverse estimates for non-local boundary integral operators, Math. Comput., 86, 308, 2651-2686 (2017) · Zbl 1368.65242
[2] Bespalov, Alex; Betcke, Timo; Haberl, Alexander; Praetorius, Dirk, Adaptive BEM with optimal convergence rates for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 346, 260-287 (2019) · Zbl 1440.65259
[3] Bartels, Sören; Carstensen, Carsten, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comput., 71, 239, 945-969 (2002) · Zbl 0997.65126
[4] Bebendorf, Mario, Hierarchical Matrices, Lecture Notes in Computational Science and Engineering., vol. 63 (2008), Springer-Verlag: Springer-Verlag Berlin · Zbl 1151.65090
[5] Börm, Steffen, Efficient Numerical Methods for Non-local Operators: \(H^2\)-Matrix Compression, Algorithms and Analysis (2010), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1208.65037
[6] Banjai, Lehel; Trefethen, Lloyd N., A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides, SIAM J. Sci. Comput., 25, 3, 1042-1065 (2003) · Zbl 1060.30012
[7] Carstensen, Carsten; Maischak, Matthias; Stephan, Ernst P., A posteriori error estimate and \(h\)-adaptive algorithm on surfaces for Symm’s integral equation, Numer. Math., 90, 2, 197-213 (2001) · Zbl 1018.65138
[8] Dörfler, Willy, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090
[9] Driscoll, Tobin A.; Trefethen, Lloyd N., Schwarz-Christoffel Mapping (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1003.30005
[10] Feischl, Michael; Führer, Thomas; Heuer, Norbert; Karkulik, Michael; Praetorius, Dirk, Adaptive boundary element methods, Arch. Comput. Methods Eng., 22, 3, 309-389 (Jul 2015) · Zbl 1348.65172
[11] Feischl, Michael; Führer, Thomas; Karkulik, Michael; Melenk, Jens Markus; Praetorius, Dirk, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation, Calcolo, 51, 4, 531-562 (2014) · Zbl 1314.65161
[12] Feischl, Michael; Führer, Thomas; Karkulik, Michael; Melenk, J. Markus; Praetorius, Dirk, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation. Part II: hyper-singular integral equation, Electron. Trans. Numer. Anal., 44, 153-176 (2015) · Zbl 1312.65173
[13] Feischl, Michael; Führer, Thomas; Karkulik, Michael; Praetorius, Dirk, ZZ-type a posteriori error estimators for adaptive boundary element methods on a curve, Eng. Anal. Bound. Elem., 38, 49-60 (2014) · Zbl 1287.65102
[14] Feischl, Michael; Gantner, Gregor; Haberl, Alexander; Praetorius, Dirk; Führer, Thomas, Adaptive boundary element methods for optimal convergence of point errors, Numer. Math., 132, 3, 541-567 (2016) · Zbl 1338.65259
[15] Führer, Thomas; Haberl, Alexander; Praetorius, Dirk; Schimanko, Stefan, Adaptive BEM with inexact PCG solver yields almost optimal computational costs, Numer. Math., 141, 967-1008 (2019) · Zbl 1412.65233
[16] Feischl, Michael; Karkulik, Michael; Melenk, Jens Markus; Praetorius, Dirk, Quasi-optimal convergence rate for an adaptive boundary element method, SIAM J. Numer. Anal., 51, 2, 1327-1348 (2013) · Zbl 1273.65186
[17] Feischl, Michael; Praetorius, Dirk; van der Zee, Kristoffer G., An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal., 54, 3, 1423-1448 (2016) · Zbl 1382.65392
[18] Gantumur, Tsogtgerel, Adaptive boundary element methods with convergence rates, Numer. Math., 124, 3, 471-516 (2013) · Zbl 1277.65038
[19] Groth, Samuel P.; Baran, Anthony J.; Betcke, Timo; Havemann, Stephan; Śmigaj, Wojciech, The boundary element method for light scattering by ice crystals and its implementation in bem++, J. Quant. Spectrosc. Radiat. Transf., 167, Supplement C, 40-52 (2015)
[20] Greengard, Leslie; Gueyffier, Denis; Martinsson, Per-Gunnar; Rokhlin, Vladimir, Fast direct solvers for integral equations in complex three-dimensional domains, Acta Numer., 18, 243-275 (2009) · Zbl 1176.65141
[21] Graham, Ivan G.; McLean, William, Anisotropic mesh refinement: the conditioning of Galerkin boundary element matrices and simple preconditioners, SIAM J. Numer. Anal., 44, 4, 1487-1513 (2006) · Zbl 1125.65111
[22] Greengard, Leslie; Rokhlin, Vladimir, A new version of the fast multipole method for the Laplace equation in three dimensions, (Acta Numerica. Acta Numerica, Acta Numer, vol. 6 (1997), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 229-269 · Zbl 0889.65115
[23] Gwinner, Joachim; Stephan, Ernst Peter, Advanced Boundary Element Methods (2018), Springer: Springer Cham · Zbl 1429.65001
[24] Hackbusch, Wolfgang, Hierarchical Matrices: Algorithms and Analysis (2015), Springer: Springer Heidelberg · Zbl 1336.65041
[25] Hiptmair, Ralf, Operator preconditioning, Comput. Math. Appl., 52, 5, 699-706 (2006) · Zbl 1125.65037
[26] Hiptmair, Ralf; Jerez-Hanckes, Carlos; Urzúa-Torres, Carolina, Mesh-independent operator preconditioning for boundary elements on open curves, SIAM J. Numer. Anal., 52, 5, 2295-2314 (2014) · Zbl 1310.65155
[27] Hwang, Chi-Ok; Mascagni, Michael; Won Monte, Taeyoung, Carlo methods for computing the capacitance of the unit cube, Math. Comput. Simul., 80, 6, 1089-1095 (2010) · Zbl 1192.78046
[28] Helsing, Johan; Perfekt, Karl-Mikael, On the polarizability and capacitance of the cube, Appl. Comput. Harmon. Anal., 34, 3, 445-468 (2013) · Zbl 1277.78016
[29] Hsiao, George C.; Wendland, Wolfgang L., Boundary Integral Equations (2008), Springer: Springer Berlin · Zbl 1157.65066
[30] Dimon Kellogg, Oliver, Foundations of Potential Theory (1967), Springer: Springer Berlin, Reprint from the first edition of 1929 · Zbl 0152.31301
[31] Karkulik, Michael; Pavlicek, David; Praetorius, Dirk, On 2D newest vertex bisection: optimality of mesh-closure and \(H^1\)-stability of \(L_2\)-projection, Constr. Approx., 38, 2, 213-234 (2013) · Zbl 1302.65267
[32] McLean, William, Strongly Elliptic Systems and Boundary Integral Equations (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0948.35001
[33] Of, Günther; Steinbach, Olaf; Wendland, Wolfgang L., The fast multipole method for the symmetric boundary integral formulation, IMA J. Numer. Anal., 26, 2, 272-296 (2006) · Zbl 1101.65114
[34] Pólya, George, Estimating electrostatic capacity, Am. Math. Mon., 54, 4, 201-206 (1947) · Zbl 0029.42102
[35] Read, Frank H., Improved extrapolation technique in the boundary element method to find the capacitances of the unit square and cube, J. Comp. Physiol., 133, 1, 1-5 (1997) · Zbl 0879.65075
[36] Rodriguez, Rodolfo, Some remarks on Zienkiewicz-Zhu estimator, Numer. Methods Partial Differ. Equ., 10, 5, 625-635 (1994) · Zbl 0806.73069
[37] Śmigaj, Wojciech; Betcke, Timo; Arridge, Simon; Phillips, Joel; Schweiger, Martin, Solving boundary integral problems with BEM++, ACM Trans. Math. Softw., 41, 2 (2015), Art. 6, 40 · Zbl 1371.65127
[38] Sauter, Stefan A.; Schwab, Christoph, Boundary Element Methods (2011), Springer: Springer Berlin · Zbl 1215.65183
[39] Stevenson, Rob, Optimality of a standard adaptive finite element method, Found. Comput. Math., 7, 2, 245-269 (2007) · Zbl 1136.65109
[40] Steinbach, Olaf, Numerical Approximation Methods for Elliptic Boundary Value Problems (2008), Springer: Springer New York · Zbl 1153.65302
[41] Steinbach, Olaf; Wendland, Wolfgang L., The construction of some efficient preconditioners in the boundary element method, Adv. Comput. Math., 9, 1-2, 191-216 (1998) · Zbl 0922.65076
[42] van ’t Wout, Elwin; Gélat, Pierre; Betcke, Timo; Arridge, Simon, A fast boundary element method for the scattering analysis of high-intensity focused ultrasound, J. Acoust. Soc. Am., 138, 5, 2726-2737 (2015)
[43] Zienkiewicz, Oleg; Zhu, Jian Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Methods Eng., 24, 2, 337-357 (1987) · Zbl 0602.73063
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