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31
Mixed Finite Element Methods for Flow in Porous Media
 RICE UNIVERSITY
, 1996
"... Mixed finite element discretizations for problems arising in flow in porous medium applications are considered. We first study second order elliptic equations which model single phase flow. We consider the recently introduced expanded mixed method. Combined with global mapping techniques, the method ..."
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Cited by 38 (18 self)
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Mixed finite element discretizations for problems arising in flow in porous medium applications are considered. We first study second order elliptic equations which model single phase flow. We consider the recently introduced expanded mixed method. Combined with global mapping techniques, the method is suitable for full conductivity tensors and general geometry domains. In the case of the lowest order RaviartThomas spaces, quadrature rules reduce the method to cellcentered finite differences, making it very efficient computationally. We consider problems with discontinuous coefficients on multiblock domains. To obtain accurate approximations, we enhance the scheme by introducing Lagrange multiplier pressures along subdomain boundaries and coefficient discontinuities. This modification comes at no extra computational cost, if the method is implemented in parallel, using nonoverlapping domain decomposition algorithms. Moreover, for regular solutions, it provides optimal convergence and...
Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals
 SIAM J. Numer. Anal
, 2004
"... Abstract. Superconvergence of the velocity is established for mimetic finite difference approximations of secondorder elliptic problems over h 2uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite d ..."
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Cited by 21 (14 self)
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Abstract. Superconvergence of the velocity is established for mimetic finite difference approximations of secondorder elliptic problems over h 2uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space. The theoretical results are confirmed by numerical experiments.
Convergence of multi point flux approximations on quadrilateral grids
 Numer. Methods Partial Differ. Equ
, 2005
"... This paper presents a convergence analysis of the multi point flux approximation control volume method, MPFA, in two space dimensions. The MPFA version discussed here is the so–called O–method on general quadrilateral grids. The discretization is based on local mappings onto a reference square. The ..."
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Cited by 18 (4 self)
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This paper presents a convergence analysis of the multi point flux approximation control volume method, MPFA, in two space dimensions. The MPFA version discussed here is the so–called O–method on general quadrilateral grids. The discretization is based on local mappings onto a reference square. The key ingredient in the analysis is an equivalence between the MPFA method and a mixed finite element method, using a specific numerical quadrature, such that the analysis of the MPFA method can be done in a finite element setting. c ○??? John Wiley &
Convergence of a Symmetric MPFA Method on Quadrilateral Grids
, 2005
"... This paper investigates different variants of the Multipoint Flux Approximation (MPFA) Omethod in 2D which rely on a transformation to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different methods appear, depending on where the ..."
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Cited by 16 (9 self)
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This paper investigates different variants of the Multipoint Flux Approximation (MPFA) Omethod in 2D which rely on a transformation to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different methods appear, depending on where the transformed permeability is evaluated. Midpoint and cornerpoint evaluations are considered. Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence of the MPFA methods is investigated numerically. For cornerpoint evaluation of the reference permeability, the same convergence behavior as the Omethod in physical space is achieved when the grids are refined uniformly or when grid perturbations of order h 2 are allowed. For h 2perturbed grids, the convergence of the normal velocities is slower for the the midpoint evaluation than for the cornerpoint evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods. The relations to the MFE methods are used to explain the loss of convergence.
Relationships among some locally conservative discretization methods which handle discontinuous coefficients
 COMPUT. GEOSCI
, 2004
"... ..."
Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data
 Numer. Math
, 1998
"... Abstract. In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and costate variables are approximated by the lowest order RaviartThomas mixed finite element s ..."
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Cited by 10 (5 self)
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Abstract. In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and costate variables are approximated by the lowest order RaviartThomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive L2 superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global L2 superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. 1.
A MULTISCALE MORTAR MULTIPOINT FLUX MIXED FINITE ELEMENT METHOD
 MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
, 1999
"... In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cellcentered finit ..."
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Cited by 7 (3 self)
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In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cellcentered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a nonoverlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method.
Quadrilatral mesh revisited
 Comput. Methods Appl. Mech. Engrg
"... Several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent. The effect of the BiSection Condition on the degenerate mesh conditions is also checked. ..."
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Cited by 5 (3 self)
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Several quadrilateral shape regular mesh conditions commonly used in the finite element method are proven to be equivalent. The effect of the BiSection Condition on the degenerate mesh conditions is also checked.
A numerical approximation of nonfickian flows with mixing length growth in porous media
 Acta Math. Univ. Comenian. (N.S
"... Abstract. The nonFickian flow of fluid in porous media is complicated by the history effect which characterizes various mixing length growth of the flow, which can be modeled by an integrodifferential equation. This paper proposes two mixed finite element methods which are employed to discretize th ..."
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Cited by 5 (3 self)
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Abstract. The nonFickian flow of fluid in porous media is complicated by the history effect which characterizes various mixing length growth of the flow, which can be modeled by an integrodifferential equation. This paper proposes two mixed finite element methods which are employed to discretize the parabolic integrodifferential equation model. An optimal order error estimate is established for one of the discretization schemes. 1.