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A multiscale mortar mixed finite element method
 Simul
, 2006
"... Abstract. We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a course grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale ..."
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Cited by 52 (14 self)
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Abstract. We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a course grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar space achieves approximation comparable to the fine scale on its coarse grid by using higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale finite element and variational multiscale methods. We derive a priori error estimates and show, with appropriate choice of the mortar space, optimal order convergence and some superconvergence on the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate subdomain and mortar grids. Numerical experiments are presented in confirmation of the theory. Key words. Multiscale, mixed finite element, mortar finite element, error estimates, a posteriori, superconvergence, multiblock, nonmatching grids AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30
A POSTERIORI ERROR ESTIMATES FOR LOWESTORDER MIXED FINITE ELEMENT DISCRETIZATIONS OF CONVECTIONDIFFUSIONREACTION EQUATIONS
, 2007
"... We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the pres ..."
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Cited by 29 (4 self)
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We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusiondispersion tensor and on a possible convection dominance. Global upper bounds for the approximation error in the energy norm are derived, where in particular all constants are evaluated explicitly, so that the estimators are fully computable. Our estimators give local lower bounds for the error as well, and they hold from the cases where convection or reaction are not present to convection or reactiondominated problems; we prove that their local efficiency depends only on local variations in the coefficients and on the local Péclet number. Moreover, the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies. The main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element, known in the mixed finite element method, and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem. Numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators.
Models, Methods and Middleware for Gridenabled Multiphysics Oil Reservoir Management
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A stochastic mortar mixed finite element method for flow in porous media with multiple rock types
 In preparation
"... Abstract. This paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcy’s law with nonstationary stochastic permeability represented as a sum of local Karhu ..."
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Cited by 4 (2 self)
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Abstract. This paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcy’s law with nonstationary stochastic permeability represented as a sum of local Karhunen–Loève expansions. The approximation uses stochastic collocation on either a tensor product or a sparse grid, coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method. The latter method requires solving a coarse scale mortar interface problem via an iterative procedure. The traditional implementation requires the solution of local fine scale linear systems on each iteration. We employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration. In the stochastic setting, the basis is further reused over multiple realizations, leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude. Error analysis and numerical experiments are presented.
Interior superconvergence in mortar mixed finite element methods on nonmatching grids
, 2008
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Contemporary Mathematics A Posteriori Error Estimates of Mixed Methods for Two Phase Flow Problems
"... Abstract. Two phase flow problem in a porous medium is governed by a system of nonlinear equations. One is an elliptic equation for the pressure and the other is a parabolic equation for the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity ..."
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Abstract. Two phase flow problem in a porous medium is governed by a system of nonlinear equations. One is an elliptic equation for the pressure and the other is a parabolic equation for the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element to approximate the pressure equation and use the standard Galerkin method to treat the concentration equation. We shall obtain an explicit a posteriori error estimator in L2 (L2) for the semidiscrete scheme of the nonlinear coupled system. 1.
for discontinuous Galerkin
, 2013
"... and discretization error estimation by equilibrated fluxes ..."
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stochastic
, 2013
"... Author manuscript, published in "accepted, Proceedings of DD21 conference. (2012)" ..."
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Author manuscript, published in "accepted, Proceedings of DD21 conference. (2012)"