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A localized orthogonal decomposition method for semi-linearelliptic problems∗∗∗

Published online by Cambridge University Press:  13 August 2014

Patrick Henning ,
Axel Målqvist  and
Daniel Peterseim
Patrick Henning
Affiliation:
Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden.
Axel Målqvist
Affiliation:
Department of Information Technology, Uppsala University, Box 337, 75105 Uppsala, Sweden.
Daniel Peterseim
Affiliation:
Institut für Numerische Simulation der Universität Bonn, Wegelerstr. 66, 53123 Bonn, Germany.. [email protected]
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Abstract

In this paper we propose and analyze a localized orthogonal decomposition (LOD) methodfor solving semi-linear elliptic problems with heterogeneous and highly variablecoefficient functions. This Galerkin-type method is based on a generalized finite elementbasis that spans a low dimensional multiscale space. The basis is assembled by performinglocalized linear fine-scale computations on small patches that have a diameter of orderH | log (H)| where H is the coarse mesh size. Without any assumptions onthe type of the oscillations in the coefficients, we give a rigorous proof for a linearconvergence of the H1-error with respect to the coarse meshsize even for rough coefficients. To solve the corresponding system of algebraicequations, we propose an algorithm that is based on a damped Newton scheme in themultiscale space.

Keywords

Finite element methoda priori error estimateconvergencemultiscale methodnon-linearcomputational homogenizationupscaling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1331 - 1349
Copyright
© EDP Sciences, SMAI 2014

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References

Alt, H.W. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311–341. Google Scholar
Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives. Pacific J. Math. 16 (1966) 1–3. Google Scholar
H. Berninger, Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation. Ph.D. thesis. Freie Universität Berlin (2007).
H. Berninger, Non-overlapping domain decomposition for the Richards equation via superposition operators. Vol. 70 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (2009) 169–176.
H. Berninger, R. Kornhuber and O. Sander, 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.
Berninger, H., Kornhuber, R. and Sander, O., Fast and robust numerical solution of the Richards equation in homogeneous soil. SIAM J. Numer. Anal. 49 (2011) 2576–2597. Google Scholar
Bourlioux, A. and Majda, A.J., An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion. Combust. Theory Model. 4 (2000) 189–210. Google Scholar
R.H. Brooks and A.T. Corey, Hydraulic properties of porous media. Hydrol. Pap. 4, Colo. State Univ., Fort Collins (1964).
Burdine, N.T., Relative permeability calculations from pore-size distribution data. Petr. Trans. Am. Inst. Mining Metall. Eng. 198 (1953) 71–77. Google Scholar
Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187–1202. Google Scholar
Carstensen, C. and Verfürth, R., Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36 (1999) 1571–1587. Google Scholar
J.E. Dennis Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM Classics Appl. Math. (1996).
E, W. and Engquist, B., The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. Google Scholar
Gloria, A., An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. SIAM Multiscale Model. Simul. 5 (2006) 996–1043. Google Scholar
Henning, P., Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Netw. Heterog. Media 7 (2012) 503–524. Google Scholar
Henning, P. and Ohlberger, M., 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. Google Scholar
Henning, P. and Ohlberger, M., A Note on Homogenization of Advection-Diffusion Problems with Large Expected Drift. Z. Anal. Anwend. 30 (2011) 319–339. Google Scholar
P. Henning and M. Ohlberger, 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).
Henning, P. and Peterseim, D., Oversampling for the Multiscale Finite Element Method. SIAM Multiscale Model. Simul. 12 (2013) 1149–1175. Google Scholar
Hou, T. and Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
Hughes, T.J.R., Feijóo, G.R., Mazzei, L. and Quincy, J.-B., The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. Google Scholar
Hughes, T.J.R. and Sangalli, G., Variational multiscale analysis: the fine-scale Green?s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45 (2007) 539–557. Google Scholar
Gardner, W.R., Some steady state solutions of unsaturated moisture ßow equations with application to evaporation from a water table. Soil Sci. 85 (1958) 228–232. Google Scholar
van Genuchten, M.T., A closedform equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44 (1980) 892–898. Google Scholar
Karátson, J., Characterizing Mesh Independent Quadratic Convergence of Newton’s Method for a Class of Elliptic Problems. J. Math. Anal. 44 (2012) 1279–1303. Google Scholar
C.T. Kelley, Iterative methods for linear and nonlinear equations. In vol. 16. SIAM Frontiers in Applied Mathematics (1996).
Larson, M.G. and Målqvist, A., 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. Google Scholar
Larson, M.G. and Målqvist, A., An adaptive variational multiscale method for convection-diffusion problems. Commun. Numer. Methods Engrg. 25 (2009) 65–79. Google Scholar
Larson, M.G. and Målqvist, A., A mixed adaptive variational multiscale method with applications in oil reservoir simulation. Math. Models Methods Appl. Sci. 19 (2009) 1017–1042. Google Scholar
Målqvist, A., Multiscale methods for elliptic problems. Multiscale Model. Simul. 9 (2011) 1064–1086. Google Scholar
A. M alqvist and D. Peterseim, Localization of Elliptic Multiscale Problems. To appear in Math. Comput. (2011). Preprint arXiv:1110.0692v4.
Mualem, Y., A New Model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media. Water Resour. Res. 12 (1976) 513–522. Google Scholar
Nordbotten, J.M., Adaptive variational multiscale methods for multiphase flow in porous media. SIAM Multiscale Model. Simul. 7 (2008) 1455–1473. Google Scholar
Peterseim, D., Robustness of Finite Element Simulations in Densely Packed Random Particle Composites. Netw. Heterog. Media 7 (2012) 113–126. Google Scholar
Peterseim, D. and Sauter, S.A., Finite Elements for Elliptic Problems with Highly Varying, Non-Periodic Diffusion Matrix. SIAM Multiscale Model. Simul. 10 (2012) 665–695. Google Scholar
M. Růžička, Nichtlineare Funktionalanalysis. Oxford Mathematical Monographs. Springer-Verlag, Berlin, Heidelberg, New York (2004).