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26 V. Girault and M.F. Wheeler [RH73] [Riv00] [RT75] W. H. Reed and T. R. Hill. Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR- 73-479, 1973. B. Rivière. Discontinuous Galerkin finite element methods for solving the miscible displacement problem in porous media. PhD thesis, University of Texas at Austin, Austin, TX, 2000. P. A. Raviart and J. M. Thomas. A mixed finite element method for second order elliptic problems. In Mathematical Aspects of Finite Element Methods, number 606 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1975. [RW74] H. Rachford and M. F. Wheeler. An H1-Galerkin procedure for the two-point boundary value problem. In C. deBoor, editor, Mathematical Aspects of Finite Elements in Partial Differential Equations, pages 353– 382. Academic Press, 1974. [RWG99] [RWG01] [SST03] [SW] B. Riviere, M. F. Wheeler, and V. Girault. Part I: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Comput. Geosci., 3:337–360, 1999. B. Riviere, M. F. Wheeler, and V. Girault. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal., 39(3):902–931, 2001. D. Shotzau, C. Schwab, and A. Toselli. Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal., 40(319):2171–2194, 2003. S. Sun and M. F. Wheeler. Discontinuous Galerkin methods for multiphase compressible flows. In preparation. [SW06a] S. Sun and M. F. Wheeler. Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport. Comput. Methods Appl. Mech. Engrg., 195(25–28):3382–3405, 2006. [SW06b] S. Sun and M. F. Wheeler. A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems. Comput. Methods Appl. Mech. Engrg., 195:632–652, 2006. [Tem79] [WD80] R. Temam. Navier–Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam, 1979. M. F. Wheeler and B. L. Darlow. Interior penalty Galerkin procedures for miscible displacement problems in porous media. In Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979), pages 485–506, Amsterdam, 1980. North- Holland. [WESR03] M. F. Wheeler, O. Eslinger, S. Sun, and B. Rivière. Discontinuous Galerkin method for modeling flow and reactive transport porous media. In Analysis and Simulation of Multifield Problems, pages 37–58. Springer-Verlag, Berlin, 2003. [Whe78] M. F. Wheeler. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal., 15(1):152–161, 1978.
Mixed Finite Element Methods on Polyhedral Meshes for Diffusion Equations Yuri A. Kuznetsov Department of Mathematics, University of Houston, 651 Philip G. Hoffman Hall, Houston, TX 77204–3008, USA kuz@math.uh.edu Summary. In this paper, a new mixed finite element method for the diffusion equation on polyhedral meshes is proposed. The method is applied to the diffusion equation on meshes with mixed cells when all the coefficients and the source function may have discontinuities inside polyhedral mesh cells. The resulting discrete equations operate only with the degrees of freedom for normal fluxes on the boundaries of cells and one degree of freedom per cell for the solution function. Key words: Diffusion equation, mixed finite element method, polyhedral meshes, mixed cells 1 Introduction In this paper, we propose a new mixed finite element method for the diffusion equation on general polyhedral meshes in the case when the coefficients of the equation and the source function may have strong discontinuities inside mesh cells. Such mesh cells are called mixed ones. The major idea of the method is reported in [Kuz05]. This work is a natural extension of the method in [Kuz06] to 3D diffusion equations. The discretization method consists of several steps. At the first step, we partition each polyhedral cell into polyhedral subcells assuming that inside each subcell the coefficients and the source function are relatively smooth. Then, in each subcell we impose a local conforming tetrahedral mesh subject to a structure of the neighboring subcells. The subcell tetrahedral meshes are not required to be conforming on the interfaces between subcells. A special finite element subspace of H div (Ω) is invented, and the classical mixed finite element method [BF91, RT91] is used for discretization of the diffusion equation with the Neumann boundary condition. At the final step, the interior (with respect to the boundaries of polyhedral mesh cells) degrees of freedom for the normal fluxes and for the solution function are eliminated, and a new
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136 S. Lapin et al. Ω R γ Fig. 3.
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140 S. Lapin et al. 5 Numerical Exp
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Domain Decomposition and Electronic
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Domain Decomposition Approach for C
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Numerical Analysis of a Finite Elem
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so that |u| 2 1,ω h ≤ 2 Numerica
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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Shape Optimization Problems with Ne
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Reduced-Order Modelling of Dispersi
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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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