Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm
Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 495-521.
Publié le :
DOI : 10.1142/S1793744211000436
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Ribot, Magali; Schatzman, Michelle. Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 495-521. doi : 10.1142/S1793744211000436. http://www.numdam.org/articles/10.1142/S1793744211000436/

[1] A. Averbuch, A. Cohen and M. Israeli, A stable and accurate explicit scheme for parabolic evolution equations, 1998, http://www.ann.jussieu.fr/∼cohen/para.ps.gz.

[2] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups gener- ated by them, Pacific J. Math. 10 (1960) 419–437.

[3] F. A. Bornemann, An adaptive multilevel approach to parabolic equations, II. Variable-order time discretization based on a multiplicative error correction, Impact Comput. Sci. Engrg. 3 (1991) 93–122.

[4] F. A. Bornemann, An adaptive multilevel approach to parabolic equations, III. 2D error estimation and multilevel preconditioning, Impact Comput. Sci. Engrg. 4 (1992) 1–45.

[5] S. H. Brill and G. F. Pinder, Analysis of a block red-black preconditioner applied to the Hermite collocation discretization of a model parabolic equation, Numer. Methods Partial Differential Equations 17 (2001) 584–606.

[6] P. N. Brown and C. S. Woodward, Preconditioning strategies for fully implicit radia- tion diffusion with material-energy transfer, SIAM J. Sci. Comput. 23 (2001) 499–516 (electronic), Copper Mountain Conference (2000).

[7] C. Canuto and A. Quarteroni, Preconditioned minimal residual methods for Cheby- shev spectral calculations, J. Comput. Phys. 60 (1985) 315–337.

[8] B. Costa, L. Dettori, D. Gottlieb and R. Temam, Time marching multilevel tech- niques for evolutionary dissipative problems, SIAM J. Sci. Comput. 23 (2001) 46–65 (electronic).

[9] M. Deville and E. Mund, Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning, J. Comput. Phys. 60 (1985) 517–533.

[10] M. O. Deville and E. H. Mund, Finite-element preconditioning for pseudospectral solutions of elliptic problems, SIAM J. Sci. Statist. Comput. 11 (1990) 311–342.

[11] B. Dia and M. Schatzman, On the order of extrapolation of integration formulae, Tech- nical Report 275, Équipe d’Analyse Numérique de Lyon, 1998, http://maply.univ- lyon1.fr/publis/publiv/1998/dia190298/texte.ps.

[12] P. Dutt, P. Biswas and G. Naga Raju, Preconditioners for spectral element methods for elliptic and parabolic problems, J. Comput. Appl. Math. 215 (2008) 152–166.

[13] I. Gerace, P. Pucci, N. Ceccarelli, M. Discepoli and R. Mariani, A preconditioned finite element method for the p-Laplacian parabolic equation, Appl. Numer. Anal. Comput. Math. 1 (2004) 155–164.

[14] P. Haldenwang, G. Labrosse, S. Abboudi and M. Deville, Chebyshev 3D spectral and 2D pseudospectral solvers for the Helmholtz equation, J. Comput. Phys. 55 (1984) 115–128.

[15] X.-Q. Jin, A note on circulant preconditioners for hyperbolic and parabolic equations, Chinese J. Math. 21 (1993) 129–142.

[16] X.-Q. Jin and R. H. Chan, Circulant preconditioners for second order hyperbolic equations, BIT 32 (1992) 650–664.

[17] T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, 1995), reprint of the 1980 edition.

[18] Y. M. Laevsky, Preconditioning operators for grid parabolic problems, Russ. J. Numer. Anal. Math. Model. 11 (1996) 497–515.

[19] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Appli- cations, Vol. I (Springer-Verlag, 1972), translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.

[20] K.-A. Mardal, T. K. Nilssen and G. A. Staff, Order-optimal preconditioners for implicit Runge-Kutta schemes applied to parabolic PDEs, SIAM J. Sci. Comput. 29 (2007) 361–375 (electronic).

[21] P. K. Moore and R. H. Dillon, A comparison of preconditioners in the solution of parabolic systems in three space dimensions using DASPK and a high order finite element method, Appl. Numer. Math. 20 (1996) 117–128, Workshop on the Method of Lines for Time-Dependent Problems, Lexington, KY, 1995.

[22] L. S. Mulholland and D. M. Sloan, The role of preconditioning in the solution of evolutionary partial differential equations by implicit Fourier pseudospectral methods, J. Comput. Appl. Math. 42 (1992) 157–174.

[23] S. A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980) 70–92.

[24] S. V. Parter, Preconditioning Legendre special collocation methods for elliptic prob- lems, I. Finite difference operators, SIAM J. Numer. Anal. 39 (2001) 330–347 (electronic).

[25] S. V. Parter, Preconditioning Legendre spectral collocation methods for elliptic problems, II. Finite element operators, SIAM J. Numer. Anal. 39 (2001) 348–362 (electronic).

[26] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Anal- ysis, Self-Adjointness (Academic Press, 1975).

[27] M. Ribot, Asymptotics of some ultra-spherical polynomials and their extrema, Methods Appl. Anal. 12 (2005) 43–74.

[28] M. Ribot, Etude théorique de méthodes numériques pour les systèmes de réaction- diffusion; application à des équations paraboliques non linéaires et non locales, Ph.D. Thesis, December 2003, http://math.unice.fr/∼ribot/recherche/these.pdf.

[29] M. Ribot and M. Schatzman, Equivalence between the spectral and the finite elements matrices, in Computational Fluid and Solid Mechanics, Proc. of the Second MIT Conference on Computational Fluid and Solid Mechanics, ed. K. J. Bathe, Cambridge, Massachusetts, USA, June 17–20, 2003 (Elsevier, 2003), http://math.unice.fr/∼ribot/recherche/RibotSchatzmanMIT2.pdf.

[30] M. Schatzman, Toward noncommutative numerical analysis: High order integra- tion in time, in Proc. of the Fifth Int. Conf. on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), Vol. 17, 2002, pp. 99–116.

[31] G. A. Staff, K.-A. Mardal and T. K. Nilssen, Preconditioning of fully implicit Runge– Kutta schemes for parabolic PDEs, Model. Identif. Control 27 (2006) 109–123.

[32] K. Yosida, Functional Analysis (Springer-Verlag, 1995), reprint of the sixth 1980 edition.

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