zbMATH — the first resource for mathematics

A domain embedding method for Dirichlet problems in arbitrary space dimension. (English) Zbl 0913.65099
Author’s abstract: An embedding method for the discretization of Dirichlet boundary value problems over general domains in arbitrary space dimension is proposed. The main advantage of the method lies in the use of Cartesian coordinates independent of the underlying domain. Error estimates and aspects of the numerical realization are considered. To obtain an efficient solver for the resulting linear system of equations an easy-to-use preconditioning is recommended and analyzed. A variety of numerical experiments illustrate and confirm the theoretical results.
Reviewer: Th.Sonar (Hamburg)

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
Full Text: DOI EuDML
[1] S. BERTOLUZZA, Interior estimates for the wavelet Galerkin method, Numer. Math. 78 (1997), pp. 1-20. Zbl0888.65113 MR1483566 · Zbl 0888.65113 · doi:10.1007/s002110050301
[2] C. BORGERS and O. B. WIDLUND, On finite element domain imbedding methods, SIAM J. Numer. Anal., 27 (1990), pp. 963-978. Zbl0705.65078 MR1051116 · Zbl 0705.65078 · doi:10.1137/0727055
[3] D. BRAESS, Finite-Elemente, Springer Lehrbuch, Springer-Verlag, Berlin, 1992. Zbl0754.65084 · Zbl 0754.65084
[4] J. H. BRAMBLE and J. E. PASCIAK, New estimates for multilevel algorithms including the V-cycle, Math. Comp., 60 (1993), pp. 447-471. Zbl0783.65081 MR1176705 · Zbl 0783.65081 · doi:10.2307/2153097
[5] J. H. BRAMBLE, J. E. PASCIAK and J. XU, Parallel multilevel preconditioners, Math. Comp., 55 (1990), pp. 1-22. Zbl0703.65076 MR1023042 · Zbl 0703.65076 · doi:10.2307/2008789
[6] C. K. CHUI, Multivariate Splines, vol. 54 of CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1988. Zbl0687.41018 MR1033490 · Zbl 0687.41018
[7] B. A. CIPRA, A rapid-deployment force for CFD : Cartesian grids, Siam News (Newsjournal of the Society for Industrial and Applied Mathematics), 25 (1995).
[8] A. COHEN, I. DAUBECHIES and J.-C. FEAUVEAU, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., 45 (1992), pp. 485-560. Zbl0776.42020 MR1162365 · Zbl 0776.42020 · doi:10.1002/cpa.3160450502
[9] S. DAHLKE, V. LATOUR and K. GRÖCHENIG, Biorthogonal box spline wavelet bases, Bericht 122, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 1995. · Zbl 0946.65149
[10] W. DAHMEN and A. KUNOTH, Multilevel preconditioning, Numer. Math., 63 (1992), pp. 315-344. Zbl0757.65031 MR1186345 · Zbl 0757.65031 · doi:10.1007/BF01385864 · eudml:133684
[11] W. DAHMEN and C. A. MICCHELLI, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal., 30 (1993), pp. 507-537. Zbl0773.65006 MR1211402 · Zbl 0773.65006 · doi:10.1137/0730024
[12] W. DAHMEN, S. PRÖSSDORF and R. SCHNEIDER, Wavelet approximation methods for pseudodifferential equations I : Stability and convergence, Math. Z., 215 (1994), pp. 583-620. Zbl0794.65082 MR1269492 · Zbl 0794.65082 · doi:10.1007/BF02571732 · eudml:174630
[13] I. DAUBECHIES, Orthonormal bases of compacity supported wavelets, Comm. Pure Appl. Math., 41 (1988), pp. 906-966. Zbl0644.42026 MR951745 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[14] C. DE BOOR, K. HÖLLING and S. RIEMENSCHNEIDER, Box Splines, vol. 98 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1993. Zbl0814.41012 MR1243635 · Zbl 0814.41012
[15] P. DEUFLHARD and A. HOHMANN, Numerical Analysis : A First Course in Scientific Computation, de Gruyter Texbook, de Gruyter, Berlin, New York, 1994. Zbl0818.65002 MR1325691 · Zbl 0818.65002
[16] G. J. FIX and G. STRANG, A Fourier analysis of the finite element method in Ritz-Galerkin theory, in Constructive Aspects of Functional Analysis, Rome, 1973, Edizioni Cremonese, pp. 265-273. Zbl0179.22501 MR258297 · Zbl 0179.22501
[17] D. GILBARG and N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der mathematischen Wissenshaften, Springer Verlag, Berlin, 1983. Zbl0562.35001 MR737190 · Zbl 0562.35001
[18] R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. Zbl0536.65054 MR737005 · Zbl 0536.65054
[19] R. GLOWINSKI and T.-W. PAN, Error estimates for fictitious domain/penalty/finite element methods, Calcolo, 19 (1992), pp. 125-141. Zbl0770.65066 MR1219625 · Zbl 0770.65066 · doi:10.1007/BF02576766
[20] R. GLOWINSKI, T.-W. PAN, R. O. Jr. WELLS and X. ZHOU, Wavelet and finite element solutions for the Neumann problem using fictitious domains, J. Comp. Phys., 126 (1996), pp. 40-51. Zbl0852.65098 MR1391621 · Zbl 0852.65098 · doi:10.1006/jcph.1996.0118
[21] R. GLOWINSKI, A. RIEDER, R. O. Jr. WELLS and X. ZHOU, A wavelet multilevel method for Dirichlet boundary value problems in general domains, Modélisation Mathématique et Analyse Numérique (M2AN), 30 (1996), pp. 711-729. Zbl0860.65121 MR1419935 · Zbl 0860.65121 · eudml:193820
[22] W. HACKBUSCH, Elliptic Differential Equations : Theory and Numerical Treatment, vol. 18 of Springer Series in Computational Mathematics, Springer Verlag, Heidelberg, 1992. Zbl0755.35021 MR1197118 · Zbl 0755.35021
[23] W. HACKBUSCH, Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1994. Zbl0789.65017 MR1247457 · Zbl 0789.65017
[24] R. H. W. HOPPE, Une méthode multigrille pour la solution des problèmes d’obstacle, Modélisation Mathématiques et Analyse Numérique (M2AN), 24 (1990), pp. 711-736. Zbl0716.65056 MR1080716 · Zbl 0716.65056 · eudml:193613
[25] S. JAFFARD, Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., 29 (1992), pp. 965-986. Zbl0761.65083 MR1173180 · Zbl 0761.65083 · doi:10.1137/0729059
[26] A. KUNOTH, Computing refinable integrals documentation of the program, Manual Institut für Geometrie und Praktische Mathemtik, RWTH Aachen, 1995.
[27] Y. A. KUZNETSOV, S. A. FINOGENOV and A. V. SUPALOV, Fictitiuos domain methods for 3D elliptic problems: algorithms and software within a parallel environment, Arbeitspapiere der GMD 726, GMD, D-53754 St. Augustin, Germany, 1993.
[28] A. LATTO, H. L. RESNIKOFF and E. TENENBAUM, The evaluation of connection coefficients of compactly supported wavelets, in Proceedings of the USA-French Workshop on Wavelets and Turbulence, Princeton University, 1991.
[29] S. V. NEPOMNYASCHIKH, Mesh theorems of traces, normalization of function traces and their inversion, Sov. J. Numer. Anal. Math. Model., 6 (1991), pp. 223-242. Zbl0816.65097 MR1126677 · Zbl 0816.65097
[30] S. V. NEPOMNYASCHIKH, Fictitious space method on unstructured grids, East-West J. Numer. Math., 3 (1995), pp. 71-79. Zbl0831.65116 MR1331485 · Zbl 0831.65116
[31] J. A. NITSCHE, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math., 11 (1968), pp. 346-348. Zbl0175.45801 MR233502 · Zbl 0175.45801 · doi:10.1007/BF02166687 · eudml:131833
[32] J. A. NITSCHE and A. H. SCHATZ, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28 (1974), pp. 937-958. Zbl0298.65071 MR373325 · Zbl 0298.65071 · doi:10.2307/2005356
[33] P. OSWALD, Multilevel Finite Element Approximation : Theory and Applications, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart, Germany, 1994. Zbl0830.65107 MR1312165 · Zbl 0830.65107
[34] G. STRANG and G. J. FIX, An Analysis of the Finite Element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, N. J., 1973. Zbl0356.65096 MR443377 · Zbl 0356.65096
[35] R. O. Jr. WELLS and X. ZHOU, Wavelet-Galerkin solutions for the Dirichlet problem, Numer. Math., 70 (1995), pp. 379-396. Zbl0824.65108 MR1330870 · Zbl 0824.65108 · doi:10.1007/s002110050125
[36] J. WLOKA, Partial Differential Equations, Cambridge University Press, Cambridge, U.K., 1987. Zbl0623.35006 MR895589 · Zbl 0623.35006
[37] J. XU, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235. Zbl0857.65129 MR1393008 · Zbl 0857.65129 · doi:10.1007/BF02238513
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.