Two-level additive Schwarz methods using rough polyharmonic splines-based coarse spaces

Abstract

This paper introduces a domain decomposition preconditioner for elliptic equations with rough coefficients. The coarse space of the domain decomposition method is constructed via the so-called rough polyharmonic splines (RPS for short). As an approximation space of the elliptic problem, RPS is known to recover the quasi-optimal convergence rate and attain the quasi-optimal localization property. The authors lay out the formulation of the RPS based domain decomposition preconditioner, and numerically verify the performance boost of this method through several examples.

Keywords

Numerical homogenization / Domain decomposition / Two-level Schwarz additive preconditioner / Rough polyharmonic splines

References

[1] Aarnes J., Hou T. Y.. Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Mathematicae Applicatae Sinica, 2002, 18(1): 63-76

[2] Atteia M.. Fonctions “spline” et noyaux reproduisants d’Aronszajn-Bergman. Rev. Française Informat, Recherche Opérationnelle, 1970, 4: 31-43

[3] Babuška I., Lipton R.. Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul., 2011, 9: 373-406

[4] Babuška I., Osborn J. E.. Can a finite element method perform arbitrarily badly?. Math. Comp., 2000, 69(230): 443-462

[5] Bensoussan A., Lions J. L., Papanicolaou G.. Asymptotic Analysis for Periodic Structure, 1978, Amsterdam: North Holland

[6] Berlyand L., Owhadi H.. Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast. Archives for Rational Mechanics and Analysis, 2010, 198(2): 677-721

[7] Duchon J.. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. RAIRO Analyse Numerique, 1976, 10: 5-12

[8] Efendiev Y., Hou T.. Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math., 2007, 57(5–7): 577-596

[9] Efendiev Y., Galvis J.. A domain decomposition preconditioner for multiscale high-contrast problems. Domain Decomposition Methods in Science and Engineering XIX, 2011, New York: Springer-Verlag 189-196

[10] Efendiev Y., Galvis J., Lazarov R., Willems J.. Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM: Mathematical Modelling and Numerical Analysis, 2012, 46(5): 1175-1199

[11] Galvis J., Efendiev Y.. Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul., 2010, 8(4): 1461-1483

[12] Galvis J., Efendiev Y.. Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Modeling Simulation, 2010, 8(5): 1621-1644

[13] De Giorgi E.. Sulla convergenza di alcune successioni di integrali del tipo dell’aera. Rendi Conti di Mat., 1975, 8: 277-294

[14] Graham I. G., Lechner P. O., Scheichl R.. Domain decomposition for multiscale PDEs. Numerische Mathematik, 2007, 106(4): 589-626

[15] Graham I. G., Scheichl R.. Robust domain decomposition algorithms for multiscale PDEs. Numerical Methods for Partial Differential Equations, 2007, 23(4): 859-878

[16] Hou T. Y., Wu X. H., Cai Z.. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp., 1999, 68(227): 913-943

[17] Hou T. Y., Wu X. H.. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 1997, 134(1): 169-189

[18] Malqvist A., Peterseim D.. Localization of elliptic multiscale problems. Math. Comp., 2014, 83: 2583-2603

[19] Mathew T. P. A.. Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Lect. Notes Comput. Sci. Eng., 2008, Berlin: Springer-Verlag

[20] Murat F., Tartar L.. Cherkarv L., Kohn R. V.. H-convergence, Séminaire d’Analyse Fonctionnelle et Numérique de l’Université. English translation: Topics in the Mathematical Modelling of Composite Materials, 1978

[21] Owhadi H., Zhang L.. Metric-based upscaling. Comm. Pure Appl. Math., 2007, 60(5): 675-723

[22] Owhadi H., Zhang L.. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. SIAM Multiscale Modeling Simulation, 2011, 9: 1373-1398

[23] Owhadi H., Zhang L., Berlyand L.. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Mathematical Modelling and Numerical Analysis, 2014, 48(2): 517-552

[24] Papanicolaou, G. C. and Varadhan, S. R. S., Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. I–II, 1979; Colloq. Math. Soc. János Bolyai, 27, North-Holland, Amsterdam, 1981, 83, 5–873.

[25] Rabut C.. Elementary m-harmonic cardinal B-splines. Numer. Algorithms, 1992, 2(1): 39-61

[26] Smith B. F.. Petter Bjørstad and William Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, 1996, Cambridge: Cambridge University Press

[27] Spagnolo S.. Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa, 1968, 22(3): 571-597

[28] Toselli A., Widlund O.. Domain Decomopsition Methods-Algorithms and Theory, 2005, Berlin: Springer-Verlag