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Houston, Paul; Schoetzau, Dominik; Wei, Xiaoxi
Publisher: Springer
Languages: English
Types: Article
We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous P^3_{k}-P_{k-1} elements whereas the magnetic part of the equations is approximated by discontinuous P^3_{k}-P_{k+1} elements. We carry out a complete a-priori error analysis and prove that the energy norm error is convergent of order O(h^k) in the mesh size h. We also show that the method is able to correctly capture and resolve the strongest magnetic singularities in non-convex polyhedral domains. These results are verified in a series of numerical experiments.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 10−2 C k h b10−4 − b k 10−6
    • [1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth domains. Math. Meth. Appl. Sci., 21:823-864, 1998.
    • [2] 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., 131:41-90, 1996.
    • [3] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39:1749-1779, 2001.
    • [4] R. Berton. Magn´etohydrodynamique. Masson, 1991.
    • [5] S.C. Brenner. Poincar´e-Friedrichs inequalities for piecewise H1−functions. SIAM J. Numer. Anal., 41:306-324, 2003.
    • [6] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. In Springer Series in Computational Mathematics, volume 15. Springer-Verlag, New York, 1991.
    • [7] F. Brezzi and M. Fortin. A minimal stabilisation procedure for mixed finite element methods. Numer. Math., 89:457-491, 2001.
    • [8] B. Cockburn, G. Kanschat, and D. Scho¨tzau. Local discontinuous Galerkin methods for the Oseen equations. Math. Comp., 73:569-593, 2004.
    • [9] B. Cockburn, G. Kanschat, and D. Scho¨tzau. A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comp., 74:1067-1095, 2005.
    • [10] B. Cockburn, G. Kanschat, and D. Scho¨tzau. A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comp., 31:61-73, 2007.
    • [11] B. Cockburn, G.E. Karniadakis, and C.-W. Shu, editors. Discontinuous Galerkin Methods. Theory, Computation and Applications, volume 11 of Lect. Notes Comput. Sci. Eng. SpringerVerlag, 2000.
    • [12] B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. J. Sci. Comp., 16:173-261, 2001.
    • [13] B. Cockburn and C.-W. Shu. Foreword for the special issue on discontinuous Galerkin methods. J. Sci. Comp., 22:1-3, 2005.
    • [14] M. Costabel and M. Dauge. Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math., 93:239-277, 2002.
    • [15] T. G. Cowling. Magnetohydrodynamics. Adam Hilger, England, 1976.
    • [16] M. Dauge. Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations. SIAM J. Math. Anal., 20:74-97, 1989.
    • [17] C. Dawson. Foreword for the special issue on discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg., 195:3, 2006.
    • [18] J.-F. Gerbeau. Probl`emes math´ematiques et num´eriques pos´es par la mod´elisation de l'´electrolyse de l'aluminium. PhD thesis, Ecole Nationale des Ponts et Chauss´ees, 1998.
    • [19] J.-F. Gerbeau. A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math., 87:83-111, 2000.
    • [20] J.-F. Gerbeau, C. Le Bris, and T. Leli`evre. Mathematical methods for the magnetohydrodynamics of liquid metals. Numerical mathematics and scientific computation. Oxford University Press, New York, 2006.
    • [21] V. Girault and P.A. Raviart. Finite element methods for Navier-Stokes equations, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1986.
    • [22] J.-L. Guermond and P. Minev. Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Num. Meth. Part. Diff. Eqs., 19:709-731, 2003.
    • [23] M.D. Gunzburger, A.J. Meir, and J.S. Peterson. On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comp., 56:523-563, 1991.
    • [24] P. Hansbo and M.G. Larson. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Engrg., 191:1895-1908, 2002.
    • [25] J.S. Hesthaven and T. Warburton. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, volume 54 of Texts in Applied Mathematics. Springer, 2008.
    • [26] R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237-339, 2002.
    • [27] P. Houston, I. Perugia, and D. Scho¨tzau. Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal., 42:434-459, 2004.
    • [28] P. Houston, I. Perugia, and D. Scho¨tzau. Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comp., 22:315-346, 2005.
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