A numerical comparison of some Multiscale Finite Element approaches for advection-dominated problems in heterogeneous media
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 851-888.

The purpose of this work is to investigate the behavior of Multiscale Finite Element type methods for advection-diffusion problems in the advection-dominated regime. We present, study and compare various options to address the issue of the simultaneous presence of both heterogeneity of scales and strong advection. Classical MsFEM methods are compared with adjusted MsFEM methods, stabilized versions of the methods, and a splitting method that treats the multiscale diffusion and the strong advection separately.

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DOI : 10.1051/m2an/2016057
Classification : 35B27, 65M60, 65M12
Mots clés : Homogenization, finite elements, highly oscillatory PDEs, advection-dominated problems
Le Bris, Claude 1 ; Legoll, Frédéric 1 ; Madiot, François 1

1 École des Ponts and INRIA, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée cedex 2, France.
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Le Bris, Claude; Legoll, Frédéric; Madiot, François. A numerical comparison of some Multiscale Finite Element approaches for advection-dominated problems in heterogeneous media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 851-888. doi : 10.1051/m2an/2016057. http://www.numdam.org/articles/10.1051/m2an/2016057/

A. Abdulle and M. Huber, Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales. Numer. Math. 126 (2014) 589–633. | DOI | MR | Zbl

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization. Multiscale Model. & Simul. 4 (2005) 790–812. | DOI | MR | Zbl

J.H. Bramble, R.D. Lazarov and J.E. Pasciak, Least-squares for second-order elliptic problems. Comput. Methods Appl. Mech. Engrg. 152 (1998) 195–210. | DOI | MR | Zbl

F. Brezzi, M.-O. Bristeau, L.P. Franca, M. Mallet and G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg. 96 (1992) 117–129. | DOI | MR | Zbl

A.N. Brooks and T. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 199–259. | DOI | MR | Zbl

E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. 35 (2013) A2752–A2780. | DOI | MR | Zbl

P. Degond, A. Lozinski, B.P. Muljadi and J. Narski, Crouzeix-Raviart MsFEM with bubble functions for diffusion and advection-diffusion in perforated media. Commun. Comput. Phys. 17 (2015) 887–907. | DOI | MR | Zbl

Y. Efendiev and T. Hou, Multiscale Finite Element Methods. Vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York (2009). | MR | Zbl

Y.R. Efendiev, T.Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37 (2000) 888–910. | DOI | MR | Zbl

A. Ern and J.-L. Guermond, Theory and practice of finite elements, vol. 159. Springer (2004). | MR | Zbl

L.P. Franca, S.L. Frey and T.J.R. Hugues, Stabilized finite element methods: I. Application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg. 95 (1992) 253–276. | DOI | MR | Zbl

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl

T. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. | DOI | MR | Zbl

T.Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. | DOI | MR | Zbl

T. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401. | DOI | MR | Zbl

T.J.R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York 1979). Vol. 34 of AMD. Amer. Soc. Mech. Engrs. (ASME), New York (1979) 19–35. | MR | Zbl

W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Vol. 33 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (2003). | MR | Zbl

J. Ku, A least-squares method for second order noncoercive elliptic partial differential equations. Math. Comput. 76 (2007) 97–114. | DOI | MR | Zbl

C. Le Bris, F. Legoll and A. Lozinski, MsFEM à la Crouzeix-Raviart for highly oscillatory elliptic problems. Chinese Ann. Math. B 34 (2013) 113–138. | DOI | MR | Zbl

C. Le Bris, F. Legoll and A. Lozinski, An MsFEM type approach for perforated domains. Multiscale Modeling & Simulation 12 (2014) 1046–1077. | DOI | MR | Zbl

C. Le Bris, F. Legoll and F. Madiot, A numerical comparison of some Multiscale Finite Element approaches for convection-dominated problems in heterogeneous media. Preprint . | arXiv | Numdam | MR

C. Le Bris, F. Legoll and F. Thomines, Multiscale Finite Element approach for weakly random problems and related issues. Math. Model. Numer. Anal. 48 (2014) 815–858. | DOI | Numdam | MR | Zbl

F. Madiot, Multiscale finite element methods for advection diffusion problems. Ph.D. thesis, Université Paris-Est (2016). Available at: http://cermics.enpc.fr/theses/2016/madiot.pdf.

F. Ouaki, Etude de schémas multi-échelles pour la simulation de réservoir. Ph.D. thesis, École Polytechnique (2013). Available at: https://tel.archives-ouvertes.fr/pastel-00922783.

P.J. Park and T. Hou, Multiscale numerical methods for singularly perturbed convection-diffusion equations. Int. J. Comput. Methods 1 (2004) 17–65. | DOI | Zbl

J. Principe and R. Codina, On the stabilization parameter in the subgrid scale approximation of scalar convection-diffusion-reaction equations on distorted meshes. Comput. Methods Appl. Mech. Engrg. 199 (2010) 1386–1402. | DOI | MR | Zbl

A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Vol. 23 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (1994). | MR | Zbl

H.-G. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems. Vol. 24 of Springer Series Comput. Math. Springer (2008). | MR | Zbl

H. Ruffieux, Multiscale finite element method for highly oscillating advection-diffusion problems in convection-dominated regime. Master’s thesis. École Polytechnique Fédérale de Lausanne, Spring 2013.

S.A. Sauter and C. Schwab, Boundary element methods, Vol. 39 of Ser. Comput. Math. Springer (2011). | MR | Zbl

W.G. Szymczak, An analysis of viscous splitting and adaptivity for steady-state convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 67 (1988) 311–354 | DOI | MR | Zbl

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