In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
@article{M2AN_2014__48_5_1331_0,
author = {Henning, Patrick and M\aa lqvist, Axel and Peterseim, Daniel},
title = {A localized orthogonal decomposition method for semi-linear elliptic problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {1331-1349},
doi = {10.1051/m2an/2013141},
mrnumber = {3264356},
zbl = {1300.35011},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2014__48_5_1331_0}
}
Henning, Patrick; Målqvist, Axel; Peterseim, Daniel. A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 1331-1349. doi : 10.1051/m2an/2013141. http://gdmltest.u-ga.fr/item/M2AN_2014__48_5_1331_0/
[1] and , Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341. | MR 706391 | Zbl 0497.35049
[2] , Minimization of functions having Lipschitz continuous first partial derivatives. Pacific J. Math. 16 (1966) 1-3. | MR 191071 | Zbl 0202.46105
[3] , Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation. Ph.D. thesis. Freie Universität Berlin (2007).
[4] , Non-overlapping domain decomposition for the Richards equation via superposition operators. Vol. 70 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (2009) 169-176. | MR 2743970 | Zbl 1284.76350
[5] , and , On nonlinear Dirichlet-Neumann algorithms for jumping nonlinearities. Domain decomposition methods in science and engineering XVI. Vol. 55 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (2007) 489-496. | MR 2334139
[6] , and , Fast and robust numerical solution of the Richards equation in homogeneous soil. SIAM J. Numer. Anal. 49 (2011) 2576-2597. | MR 2873248 | Zbl 1298.76115
[7] and , An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion. Combust. Theory Model. 4 (2000) 189-210. | MR 1764195 | Zbl 1112.80308
[8] and , Hydraulic properties of porous media. Hydrol. Pap. 4, Colo. State Univ., Fort Collins (1964).
[9] , Relative permeability calculations from pore-size distribution data. Petr. Trans. Am. Inst. Mining Metall. Eng. 198 (1953) 71-77.
[10] , Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187-1202. | Numdam | MR 1736895 | Zbl 0948.65113
[11] and , Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 (1999) 1571-1587. | MR 1706735 | Zbl 0938.65124
[12] and , Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM Classics Appl. Math. (1996). | MR 1376139 | Zbl 0847.65038
[13] W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87-132. | MR 1979846 | Zbl 1093.35012
[14] , An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. SIAM Multiscale Model. Simul. 5 (2006) 996-1043. | MR 2272308 | Zbl 1119.74038
[15] , Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Netw. Heterog. Media 7 (2012) 503-524. | MR 2982460 | Zbl 1263.35074
[16] and , The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Netw. Heterog. Media 5 (2010) 711-744. | MR 2740530 | Zbl 1264.65161
[17] and , A Note on Homogenization of Advection-Diffusion Problems with Large Expected Drift. Z. Anal. Anwend. 30 (2011) 319-339. | MR 2819498 | Zbl 1223.35042
[18] and , Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Preprint 01/11 - N, to appear in DCDS-S, special issue on Numerical Methods based on Homogenization and Two-Scale Convergence (2011). | MR 3286910 | Zbl pre06377076
[19] and , Oversampling for the Multiscale Finite Element Method. SIAM Multiscale Model. Simul. 12 (2013) 1149-1175. | MR 3123820 | Zbl 1297.65155
[20] and , A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR 1455261 | Zbl 0880.73065
[21] , , and , The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR 1660141 | Zbl 1017.65525
[22] and , Variational multiscale analysis: the fine-scale Green?s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539-557. | MR 2300286 | Zbl 1152.65111
[23] , Some steady state solutions of unsaturated moisture ßow equations with application to evaporation from a water table. Soil Sci. 85 (1958) 228-232.
[24] , A closedform equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44 (1980) 892-898.
[25] , Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems. J. Math. Anal. 44 (2012) 1279-1303. | MR 2982712 | Zbl 1252.65189
[26] , Iterative methods for linear and nonlinear equations. In vol. 16. SIAM Frontiers in Applied Mathematics (1996). | MR 1344684 | Zbl 0832.65046
[27] and , Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2313-2324. | MR 2319044 | Zbl 1173.74431
[28] and , An adaptive variational multiscale method for convection-diffusion problems. Commun. Numer. Methods Engrg. 25 (2009) 65-79. | MR 2484188 | Zbl 1156.76047
[29] and , A mixed adaptive variational multiscale method with applications in oil reservoir simulation. Math. Models Methods Appl. Sci. 19 (2009) 1017-1042. | MR 2553176 | Zbl 1257.65068
[30] , Multiscale methods for elliptic problems. Multiscale Model. Simul. 9 (2011) 1064-1086. | MR 2831590 | Zbl 1248.65124
[31] A. M alqvist and D. Peterseim, Localization of Elliptic Multiscale Problems. To appear in Math. Comput. (2011). Preprint arXiv:1110.0692v4. | Zbl 1301.65123
[32] , A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media. Water Resour. Res. 12 (1976) 513-522.
[33] , Adaptive variational multiscale methods for multiphase flow in porous media. SIAM Multiscale Model. Simul. 7 (2008) 1455-1473. | MR 2496709 | Zbl 1172.76041
[34] , Robustness of Finite Element Simulations in Densely Packed Random Particle Composites. Netw. Heterog. Media 7 (2012) 113-126. | MR 2908612 | Zbl 1262.35010
[35] and , Finite Elements for Elliptic Problems with Highly Varying, Non-Periodic Diffusion Matrix. SIAM Multiscale Model. Simul. 10 (2012) 665-695. | MR 3022017 | Zbl 1264.65195
[36] , Nichtlineare Funktionalanalysis. Oxford Mathematical Monographs. Springer-Verlag, Berlin, Heidelberg, New York (2004).