A domain embedding method for Dirichlet problems in arbitrary space dimension

Rieder, Andreas

ESAIM: Modélisation mathématique et analyse numérique, Tome 32 (1998) no. 4, pp. 405-431.

[1] S. Bertoluzza, Interior estimates for the wavelet Galerkin method, Numer. Math. 78 (1997), pp. 1-20. | MR | Zbl

[2] C. Borgers and O. B. Widlund, On finite element domain imbedding methods, SIAM J. Numer. Anal., 27 (1990), pp. 963-978. | MR | Zbl

[3] D. Braess, Finite-Elemente, Springer Lehrbuch, Springer-Verlag, Berlin, 1992. | Zbl

[4] J. H. Bramble and J. E. Pasciak, New estimates for multilevel algorithms including the V-cycle, Math. Comp., 60 (1993), pp. 447-471. | MR | Zbl

[5] J. H. Bramble, J. E. Pasciak and J. Xu, Parallel multilevel preconditioners, Math. Comp., 55 (1990), pp. 1-22. | MR | Zbl

[6] C. K. Chui, Multivariate Splines, vol. 54 of CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1988. | MR | Zbl

[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. | MR | Zbl

[9] S. Dahlke, V. Latour and K. Gröchenig, Biorthogonal box spline wavelet bases, Bericht 122, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, 1995.

[10] W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math., 63 (1992), pp. 315-344. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[13] I. Daubechies, Orthonormal bases of compacity supported wavelets, Comm. Pure Appl. Math., 41 (1988), pp. 906-966. | MR | Zbl

[14] C. De Boor, K. Hölling and S. Riemenschneider, Box Splines, vol. 98 of Applied Mathematical Sciences, Springer-Verlag, Berlin, 1993. | MR | Zbl

[15] P. Deuflhard and A. Hohmann, Numerical Analysis : A First Course in Scientific Computation, de Gruyter Texbook, de Gruyter, Berlin, New York, 1994. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. | MR | Zbl

[19] R. Glowinski and T.-W. Pan, Error estimates for fictitious domain/penalty/finite element methods, Calcolo, 19 (1992), pp. 125-141. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[22] W. Hackbusch, Elliptic Differential Equations : Theory and Numerical Treatment, vol. 18 of Springer Series in Computational Mathematics, Springer Verlag, Heidelberg, 1992. | MR | Zbl

[23] W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1994. | MR | Zbl

[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. | Numdam | MR | Zbl

[25] S. Jaffard, Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., 29 (1992), pp. 965-986. | MR | Zbl

[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. | MR | Zbl

[30] S. V. Nepomnyaschikh, Fictitious space method on unstructured grids, East-West J. Numer. Math., 3 (1995), pp. 71-79. | MR | Zbl

[31] J. A. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math., 11 (1968), pp. 346-348. | MR | Zbl

[32] J. A. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp., 28 (1974), pp. 937-958. | MR | Zbl

[33] P. Oswald, Multilevel Finite Element Approximation : Theory and Applications, Teubner Skripten zur Numerik, B. G. Teubner, Stuttgart, Germany, 1994. | MR | Zbl

[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. | MR | Zbl

[35] R. O. Jr. Wells and X. Zhou, Wavelet-Galerkin solutions for the Dirichlet problem, Numer. Math., 70 (1995), pp. 379-396. | MR | Zbl

[36] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, U.K., 1987. | MR | Zbl

[37] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235. | MR | Zbl