Efficient preconditioning by the domain decomposition method for the $p$-version with a~hierarchical basis.~I

V. G. Korneev ; S. Ensen

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (1999), pp. 37-56.

Bibliographie

[1] Babuška I., Craig A., Mandel J., Pitkäranta J., “Efficient preconditioning for the $p$-version finite element method in two dimensions”, SIAM J. Numer. Anal., 28:3 (1991), 624–661 | DOI | MR | Zbl

[2] Babuška I., Griebel M., Pitkäranta J., “The problem of selecting the shape functions for a $p$-type finite element”, J. Numer. Meth. Engrg., 28 (1989), 1891–1908 | DOI | MR | Zbl

[3] Bernardi C., Maday Y., “Polynomial interpolation results in Sobolev spaces”, J. Comput. and Appl. Math., 43 (1992), 53–80 | DOI | MR | Zbl

[4] Bramble J., Pasciak J., Schatz A., “The construction of preconditioners for elliptic problems by substructuring, I”, Math. Comp., 47 (1986), 1–16 | DOI | MR

[5] Bramble J., Pasciak J., Schatz A., “The construction of preconditioners for elliptic problems by substructuring, II”, Math. Comp., 49 (1987), 415–430 | MR

[6] Bramble J., Pasciak J., Schatz A., “The construction of preconditioners for elliptic problems by substructuring, III”, Math. Comp., 51 (1988), 1–24 | DOI | MR

[7] Bramble J., Pasciak J., Schatz A., “The construction of preconditioners for elliptic problems by substructuring, IV”, Math. Comp., 53 (1989), 103–134 | DOI | MR

[8] Canuto C., “Stabilization of spectral methods by finite element bubble functions”, Comput. Meth. Appl. Mech. and Engrg., 116 (1994), 13–26 | DOI | MR | Zbl

[9] Deville M. O., Mund E. H., “Finite element preconditioning for pseudospectral solutions of elliptic problems”, SIAM J. Sci. Stat. Comput., 11:2 (1990), 311–342 | DOI | MR | Zbl

[10] Deville M. O., Mund E. H., Van Keminade V., “Preconditioned Chebyshev collocation methods and triangular finite elements”, Comput. Meth. Appl. Mech. and Engrg., 116 (1994), 193–200 | DOI | MR | Zbl

[11] Han W., Jensen S., “On the sharpness of $L^2$-error estimates of $H_0^1$-projections onto subspaces of piecewise, high order polynomials”, Math. Comput., 64 (1995), 51–70 | DOI | MR | Zbl

[12] Ivanov S. A., Korneev V. G., “Postroenie koordinatnykh funktsii vysokogo poryadka i predobuslavlivanie v ramkakh metoda dekompozitsii oblasti”, Izv. vuzov. Matematika, 1995, no. 4, 62–81 | MR | Zbl

[13] Ivanov S. A., Korneev V. G., On the preconditioning in the domain decomposition technique for the $p$-version finite element method, Part I, Preprint SPC 95-35, No 12, Technische Universität Chemnitz-Zwickau, 1995, P. 1–15

[14] Ivanov S. A., Korneev V. G., On the preconditioning in the domain decomposition technique for the $p$-version finite element method, Part II, Preprint SPC 95-36, No 12, Technische Universität Chemnitz-Zwickau, 1995, P. 1–14

[15] Ivanov S. A., Korneev V. G., “Preconditioning in the domain decomposition methods for the $p$-version with the hierarchical bases”, Proc. of OFEA-95 (St. Petersburg); Math. Modeling, 8:9 (1996), 63–73 | Zbl

[16] Jensen S., “$p$-version of mixed finite element methods for Stokes-like problems”, Comput. Meth. Appl. Mech. and Engrg., 201 (1992), 27–41 | DOI | MR

[17] Maday Y., Meiron D., Patera A. T., Rönquist E. M., “Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretization”, SIAM J. Sci. Comput., 14 (1993), 310–337 | DOI | MR | Zbl

[18] Maday Y., Patera A. T., “Spectral element methods for Navier-Stokes equations”, A. Noor, J. T. Oden, State of Art Surveys in Computational Mechanics, ASME, New York, 1996

[19] Oden J. T., Patra Abani, Feng Y. S., “Domain decomposition solver for adaptive hp finite elements”, VII-th International Conference on Domain Decomposition (State College, Pennsylvania, October, 1993)

[20] Parter S. V., Rothman E. E., “Preconditioning Legendre spectral collocation approximations to elliptic problems”, SIAM J. Numer. Anal., 32:2 (1995), 333–385 | DOI | MR | Zbl

[21] Pavarino L. F., “Additive Schwartz methods for the $p$-version finite element method”, Numer. Math., 66:4 (1994), 493–515 | MR | Zbl

[22] Pavarino L. F., Widlund O. B., “Iterative substructuring methods for spectral elements in three dimensions”, The Finite Element Meth.: Fifty Years of the Courant Element, Proc. of FEM50, eds. M. Krizek, P. Neitaanmaki, Marcell Dekker, New York–Basel–Hong Kong, 1994, 345–355 | MR | Zbl

[23] Amon C. N., “Spectral element — Fourier approximation for the Navier-Stokes equations some aspects of parallel implementation”, Parallel and Vector computations in Heat Transfer, HTD, 133, eds. J. D. Georgiadis, J. Y. Murphy, 1990

[24] Carey G. F., Barragy E., “Basis function selection and preconditioning high degree finite element and spectral methods”, BIT, 29:4 (1989), 784–804 | DOI | MR

[25] Fisher P. F., “Spectral element solution for the Navier-Stokes equations on high performance distributed memory parallel processors”, Parallel and Vector Computations in Heat Transfer, HTD, 133, eds. J. D. Georgiadis, J. Y. Murphy, 1990

[26] Mandel J., “Iterative solvers by substructuring for the $p$-version finite element method”, Comput. Meth. Appl. Mech. Engrg., 80 (1990), 117–128 | DOI | MR | Zbl

[27] Oden J. T., “Theory and implementation of high-order adaptive $h$-$p$ methods for the analysis of incompressible viscous flows”, Comput. non linear mech. in aerospace eng., AIAA progress in aeronautics, ed. S. N. Atlyri, 1991

[28] Dryja M., “A finite element capacitance method for elliptic problems on regions partitioned into substructures”, Numer. Math., 44 (1984), 153–168 | DOI | MR | Zbl

[29] Adams R. A., Sobolev Spaces, Academic Press, New York, 1975, 268 pp. | MR | Zbl

[30] Korneev V. G., “O svyazi mezhdu metodom konechnykh raznostei i metodom funktsii Grina s ispolzovaniem kvadraturnykh formul dlya obyknovennykh differentsialnykh operatorov vtorogo poryadka”, Vestn. Leningr. un-ta, 1972, no. 7, 36–44 | MR | Zbl

[31] Maitre J. F., Pourquier O., “Estimates of $\mathcal{O}(p^\alpha)$ type for the condition number of matrices in the $p$-version of the finite element method”, Proc. of Third International Conference on Spectral and Higher Order Meth. (1996); Houston J. Math., 1996, 215–220

[32] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, 4-e izd., Fizmatgiz, M., 1962, 1100 pp. | MR

[33] Ainsworth M., “A preconditioner based on domain decomposition for $h$-$p$ finite element approximation on quasi-uniform meshes”, SIAM J. Numer. Anal., 33:4 (1996), 1358–1376 | DOI | MR | Zbl

[34] Mandel J., “Hierarchical preconditioning and partial orthogonalization for the $p$-version finite element method”, Proc. of Third International Symposium on Domain Decomposition Meth. for Partial Differential Equations, eds. T. F. Chan et al., SIAM, 1989 | MR

[35] Korneev V. G., “O postroenii variatsionno-raznostnykh skhem vysokikh poryadkov tochnosti”, Vestn. Leningr. un-ta, 1970, no. 19, 28–40 | MR | Zbl

[36] Korneev V. G., Skhemy metoda konechnykh elementov vysokikh poryadkov tochnosti, Izd-vo Leningr. un-ta, L., 1977, 205 pp. | MR

[37] Korneev V. G., “Iteratsionnye metody resheniya konechnoelementnykh sistem lineinykh algebraicheskikh uravnenii”, Zhurn. vychisl. matem. i matem. fiz., 17:5 (1977), 1213–1233 | MR | Zbl