Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations

Baňas, Ľubomír; Prohl, Andreas

doi:10.1090/S0025-5718-10-02341-0

Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations
HTML articles powered by AMS MathViewer

by Ľubomír Baňas and Andreas Prohl PDF

Math. Comp. 79 (2010), 1957-1999 Request permission

Abstract:

We propose a convergent implicit stabilized finite element discretization of the nonstationary incompressible magnetohydrodynamics equations with variable density, viscosity, and electric conductivity. The discretization satisfies a discrete energy law, and a discrete maximum principle for the positive density, and iterates converge to weak solutions of the limiting problem for vanishing discretization parameters. A simple fixed point scheme, together with an appropriate stopping criterion is proposed, which decouples the computation of density, velocity, and magnetic field, and inherits the above properties, provided a mild mesh constraint holds. Computational studies are provided.

References

C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
F. Armero and J. C. Simo, Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 131 (1996), no. 1-2, 41–90. MR 1393572, DOI 10.1016/0045-7825(95)00931-0
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Zhiming Chen, Qiang Du, and Jun Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000), no. 5, 1542–1570. MR 1759906, DOI 10.1137/S0036142998349977
J. F. Ciavaldini, Analyse numerique d’un problème de Stefan à deux phases par une methode d’éléments finis, SIAM J. Numer. Anal. 12 (1975), 464–487 (French, with English summary). MR 391741, DOI 10.1137/0712037
Martin Costabel and Monique Dauge, Weighted regularization of Maxwell equations in polyhedral domains. A rehabilitation of nodal finite elements, Numer. Math. 93 (2002), no. 2, 239–277. MR 1941397, DOI 10.1007/s002110100388
M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
J.-F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numer. Math. 87 (2000), no. 1, 83–111. MR 1800155, DOI 10.1007/s002110000193
Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre, Mathematical methods for the magnetohydrodynamics of liquid metals, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006. MR 2289481, DOI 10.1093/acprof:oso/9780198566656.001.0001
V. Girault, R. H. Nochetto, and R. Scott, Maximum-norm stability of the finite element Stokes projection, J. Math. Pures Appl. (9) 84 (2005), no. 3, 279–330 (English, with English and French summaries). MR 2121575, DOI 10.1016/j.matpur.2004.09.017
J.-F. Gerbeau, T. Lelièvre, and C. Le Bris, Simulations of MHD flows with moving interfaces, J. Comput. Phys. 184 (2003), no. 1, 163–191. MR 1961974, DOI 10.1016/S0021-9991(02)00025-6
D. T. Graves, D. Trebotich, G. H. Miller, and P. Colella, An efficient solver for the equations of resistive MHD with spatially-varying resistivity, J. Comput. Phys. 227 (2008), no. 10, 4797–4804. MR 2414835, DOI 10.1016/j.jcp.2008.01.044
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Max D. Gunzburger, Amnon J. Meir, and Janet S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comp. 56 (1991), no. 194, 523–563. MR 1066834, DOI 10.1090/S0025-5718-1991-1066834-0
Urs Hasler, Anna Schneebeli, and Dominik Schötzau, Mixed finite element approximation of incompressible MHD problems based on weighted regularization, Appl. Numer. Math. 51 (2004), no. 1, 19–45. MR 2083323, DOI 10.1016/j.apnum.2004.02.005
John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311. MR 650052, DOI 10.1137/0719018
Wolfgang Hackbusch, Multigrid methods and applications, Springer Series in Computational Mathematics, vol. 4, Springer-Verlag, Berlin, 1985. MR 814495, DOI 10.1007/978-3-662-02427-0
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
Volker John, Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations, Internat. J. Numer. Methods Fluids 40 (2002), no. 6, 775–798. MR 1928952, DOI 10.1002/fld.377
Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
Sergey Korotov and Michal Křížek, Acute type refinements of tetrahedral partitions of polyhedral domains, SIAM J. Numer. Anal. 39 (2001), no. 2, 724–733. MR 1860255, DOI 10.1137/S003614290037040X
D. Kuzmin, On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection, J. Comput. Phys. 219 (2006), no. 2, 513–531. MR 2274948, DOI 10.1016/j.jcp.2006.03.034
Dmitri Kuzmin and Matthias Möller, Algebraic flux correction. I. Scalar conservation laws, Flux-corrected transport, Sci. Comput., Springer, Berlin, 2005, pp. 155–206. MR 2129255, DOI 10.1007/3-540-27206-2_{6}
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
J.L. Lions, E. Magenes, Nonhonmogeneous boundary value problems and applications, Volume I, Springer-Verlag, New York (1972).
Chun Liu and Noel J. Walkington, Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM J. Numer. Anal. 45 (2007), no. 3, 1287–1304. MR 2318813, DOI 10.1137/050629008
Peter Monk, Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447, DOI 10.1093/acprof:oso/9780198508885.001.0001
David Munger and Alain Vincent, A level set approach to simulate magnetohydrodynamic-instabilities in aluminum reduction cells, J. Comput. Phys. 217 (2006), no. 2, 295–311. MR 2260603, DOI 10.1016/j.jcp.2006.01.002
Andreas Prohl, Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system, M2AN Math. Model. Numer. Anal. 42 (2008), no. 6, 1065–1087. MR 2473320, DOI 10.1051/m2an:2008034
A. Schmidt and K. G. Siebert, ALBERT—software for scientific computations and applications, Acta Math. Univ. Comenian. (N.S.) 70 (2000), no. 1, 105–122. MR 1865363
R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. MR 1422252, DOI 10.1090/surv/049
Dominik Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Numer. Math. 96 (2004), no. 4, 771–800. MR 2036365, DOI 10.1007/s00211-003-0487-4
Jacques Simon, Sobolev, Besov and Nikol′skiĭ fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl. (4) 157 (1990), 117–148. MR 1108473, DOI 10.1007/BF01765315
Noel J. Walkington, Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. Numer. Anal. 42 (2005), no. 5, 1801–1817. MR 2139223, DOI 10.1137/S0036142902412233
Jun Zhao, Analysis of finite element approximation for time-dependent Maxwell problems, Math. Comp. 73 (2004), no. 247, 1089–1105. MR 2047079, DOI 10.1090/S0025-5718-03-01603-X