Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales

Abdulle, Assyr

doi:10.1090/S0025-5718-2011-02527-5

Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales
HTML articles powered by AMS MathViewer

by Assyr Abdulle PDF

Math. Comp. 81 (2012), 687-713 Request permission

Abstract:

An analysis of a multiscale symmetric interior penalty discontinuous Galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed. This new method, first described in [A. Abdulle, C.R. Acad. Sci. Paris, Ser. I 346 (2008)] is based on numerical homogenization. It allows to significantly reduce the computational cost of a fine scale discontinuous Galerkin method by probing the fine scale data on sampling domains within a macroscopic partition of the computational domain. Macroscopic numerical fluxes, an essential ingredient of discontinuous Galerkin finite elements, can be recovered from the computation on the sampling domains with negligible computation overhead. Fully discrete a priori error bounds are derived in the $L^2$ and $H^1$ norms.

References

Jørg Aarnes and Bjørn-Ove Heimsund, Multiscale discontinuous Galerkin methods for elliptic problems with multiple scales, Multiscale methods in science and engineering, Lect. Notes Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005, pp. 1–20. MR 2161705, DOI 10.1007/3-540-26444-2_{1}
Assyr Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput. 23 (2002), no. 6, 2041–2054. MR 1923724, DOI 10.1137/S1064827500379549
Assyr Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul. 4 (2005), no. 2, 447–459. MR 2162863, DOI 10.1137/040607137
Assyr Abdulle, Multiscale methods for advection-diffusion problems, Discrete Contin. Dyn. Syst. suppl. (2005), 11–21. MR 2192655
Assyr Abdulle, Analysis of a heterogeneous multiscale FEM for problems in elasticity, Math. Models Methods Appl. Sci. 16 (2006), no. 4, 615–635. MR 2218216, DOI 10.1142/S0218202506001285
A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs, Multiple scales problems in biomathematics, mechanics, physics and numerics, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 31, Gakk\B{o}tosho, Tokyo, 2009, pp. 133–181. MR 2590959
A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: a unified framework, to appear in Ser. Contemp. Appl. Math., CAM, World Sci. Publishing, Singapore.
Assyr Abdulle and Alexei A. Medovikov, Second order Chebyshev methods based on orthogonal polynomials, Numer. Math. 90 (2001), no. 1, 1–18. MR 1868760, DOI 10.1007/s002110100292
Assyr Abdulle and Christoph Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces, Multiscale Model. Simul. 3 (2004/05), no. 1, 195–220. MR 2123116, DOI 10.1137/030600771
Assyr Abdulle and Bjorn Engquist, Finite element heterogeneous multiscale methods with near optimal computational complexity, Multiscale Model. Simul. 6 (2007/08), no. 4, 1059–1084. MR 2393025, DOI 10.1137/060676118
Assyr Abdulle, Multiscale method based on discontinuous Galerkin methods for homogenization problems, C. R. Math. Acad. Sci. Paris 346 (2008), no. 1-2, 97–102 (English, with English and French summaries). MR 2385064, DOI 10.1016/j.crma.2007.11.029
Assyr Abdulle and Achim Nonnenmacher, A short and versatile finite element multiscale code for homogenization problems, Comput. Methods Appl. Mech. Engrg. 198 (2009), no. 37-40, 2839–2859. MR 2567848, DOI 10.1016/j.cma.2009.03.019
Assyr Abdulle and Achim Nonnenmacher, A posteriori error analysis of the heterogeneous multiscale method for homogenization problems, C. R. Math. Acad. Sci. Paris 347 (2009), no. 17-18, 1081–1086 (English, with English and French summaries). MR 2554581, DOI 10.1016/j.crma.2009.07.004
A. Abdulle and G. Vilmart, Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: a fully discrete space-time analysis, preprint submitted for publication.
Grégoire Allaire and Robert Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul. 4 (2005), no. 3, 790–812. MR 2203941, DOI 10.1137/040611239
Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. MR 1885715, DOI 10.1137/S0036142901384162
I. Babuška and J. E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20 (1983), no. 3, 510–536. MR 701094, DOI 10.1137/0720034
Ivo Babuška, Gabriel Caloz, and John E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), no. 4, 945–981. MR 1286212, DOI 10.1137/0731051
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, Vol. III, Interscience Publishers, a division of John Wiley & Sons, Inc., New York-London-Sydney, 1964. With special lectures by Lars Garding and A. N. Milgram. MR 0163043
Alain Bourgeat and Andrey Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 2, 153–165 (English, with English and French summaries). MR 2044813, DOI 10.1016/S0246-0203(03)00065-7
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
Shanqin Chen, Weinan E, and Chi-Wang Shu, The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems, Multiscale Model. Simul. 3 (2005), no. 4, 871–894. MR 2164241, DOI 10.1137/040612622
Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu, The development of discontinuous Galerkin methods, Discontinuous Galerkin methods (Newport, RI, 1999) Lect. Notes Comput. Sci. Eng., vol. 11, Springer, Berlin, 2000, pp. 3–50. MR 1842161, DOI 10.1007/978-3-642-59721-3_{1}
Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4) 8 (1973), 391–411 (Italian, with English summary). MR 0348255
Rui Du and Pingbing Ming, Heterogeneous multiscale finite element method with novel numerical integration schemes, Commun. Math. Sci. 8 (2010), no. 4, 863–885. MR 2744910
Maksymilian Dryja, On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, Comput. Methods Appl. Math. 3 (2003), no. 1, 76–85. Dedicated to Raytcho Lazarov. MR 2002258
Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. MR 1979846
Weinan E, Pingbing Ming, and Pingwen Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156. MR 2114818, DOI 10.1090/S0894-0347-04-00469-2
Yekaterina Epshteyn and Béatrice Rivière, Estimation of penalty parameters for symmetric interior penalty Galerkin methods, J. Comput. Appl. Math. 206 (2007), no. 2, 843–872. MR 2333718, DOI 10.1016/j.cam.2006.08.029
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
Thomas Y. Hou, Xiao-Hui Wu, and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68 (1999), no. 227, 913–943. MR 1642758, DOI 10.1090/S0025-5718-99-01077-7
Yalchin Efendiev and Thomas Y. Hou, Multiscale finite element methods, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4, Springer, New York, 2009. Theory and applications. MR 2477579
Viet Ha Hoang and Christoph Schwab, High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul. 3 (2004/05), no. 1, 168–194. MR 2123115, DOI 10.1137/030601077
Paul Houston, Christoph Schwab, and Endre Süli, Discontinuous $hp$-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal. 39 (2002), no. 6, 2133–2163. MR 1897953, DOI 10.1137/S0036142900374111
V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546, DOI 10.1007/978-3-642-84659-5
Ioannis G. Kevrekidis, C. William Gear, James M. Hyman, Panagiotis G. Kevrekidis, Olof Runborg, and Constantinos Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci. 1 (2003), no. 4, 715–762. MR 2041455
O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985. Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. MR 793735, DOI 10.1007/978-1-4757-4317-3
Ana-Maria Matache and Christoph Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal. 36 (2002), no. 4, 537–572. MR 1932304, DOI 10.1051/m2an:2002025
François Murat and Luc Tartar, $H$-convergence, Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, Birkhäuser Boston, Boston, MA, 1997, pp. 21–43. MR 1493039
J. Tinsley Oden and Kumar S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms, J. Comput. Phys. 164 (2000), no. 1, 22–47. MR 1786241, DOI 10.1006/jcph.2000.6585
Giancarlo Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul. 1 (2003), no. 3, 485–503. MR 2030161, DOI 10.1137/S1540345902411402
Hans Joachim Schmid, On cubature formulae with a minimum number of knots, Numer. Math. 31 (1978/79), no. 3, 281–297. MR 514598, DOI 10.1007/BF01397880
E.B. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Phil. Mag., A73 (1996), 1529–1563.
K. Terada and N. Kikuchi, A class of general algorithms for multi-scale analyses of heterogeneous media, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 40-41, 5427–5464. MR 1843662, DOI 10.1016/S0045-7825(01)00179-7
Mary Fanett Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152–161. MR 471383, DOI 10.1137/0715010