Results 1  10
of
25
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 ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
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.
New perspectives on polygonal and polyhedral finite element methods
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2014
"... Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the Virtual Element Method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more indepth understanding of mimetic schemes, and also endows polygonalbased Galerkin methods with greater flexibility than threenode and fournode finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semidefinite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinatebased Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates postprocessing of field variables and
Accurate modelling of faults by multipoint, mimetic, and mixed methods. Preprint 2010. http://www.sintef.no/Geoscale/ Created with LATEX beamerposter http://www.sintef.no/MRST Bard.Skaflestad@sintef.no
"... Abstract. Traditional flow simulators based on twopoint discretizations are not able to incorporate fault geometry and transmissibilities in a satisfactory way. Modellers are forced to either make errors by adapting faults to a grid that is almost Korthogonal or to adapt the grid to the faults an ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Traditional flow simulators based on twopoint discretizations are not able to incorporate fault geometry and transmissibilities in a satisfactory way. Modellers are forced to either make errors by adapting faults to a grid that is almost Korthogonal or to adapt the grid to the faults and hence introduce discretization errors because of the lack of Korthogonality. We propose a new method based on a hybridized mixed or mimetic discretization, which also includes the MPFAO method. The new method represents faults as internal boundaries and calculates fault transmissibilities directly instead of using multipliers to modify griddependent transmissibilities. The resulting method is geologydriven and consistent for cells with planar surfaces and thereby avoids the grid errors inherent in the twopoint method. We also propose a method to translate fault transmissibility multipliers into fault transmissibilities. This makes the new method readily applicable to reservoir models that contain fault multipliers. 1.
FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS: INTRODUCTION TO AND REVIEW OF MODERN METHODS
, 2013
"... ..."
Discontinuous Galerkin and Mimetic Finite Difference Methods for Coupled StokesDarcy Flows on Polygonal and Polyhedral Grids
, 2010
"... We study locally mass conservative approximations of coupled Darcy and Stokes flows on polygonal and polyhedral meshes. The discontinuous Galerkin (DG) finite element method is used in the Stokes region and the mimetic finite difference method is used in the Darcy region. DG finite element spaces ar ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We study locally mass conservative approximations of coupled Darcy and Stokes flows on polygonal and polyhedral meshes. The discontinuous Galerkin (DG) finite element method is used in the Stokes region and the mimetic finite difference method is used in the Darcy region. DG finite element spaces are defined on polygonal and polyhedral grids by introducing lifting operators mapping mimetic degrees of freedom to functional spaces. Optimal convergence estimates for the numerical scheme are derived. Results from computational experiments supporting the theory are presented. 1
HIERARCHICAL A POSTERIORI ERROR ESTIMATORS FOR THE MIMETIC DISCRETIZATION OF ELLIPTIC PROBLEMS
"... Abstract. We present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed a posteriori estimator are proved. Several numerical experiments assess the actual perfo ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We present an a posteriori error estimate of hierarchical type for the mimetic discretization of elliptic problems. Under a saturation assumption, the global reliability and efficiency of the proposed a posteriori estimator are proved. Several numerical experiments assess the actual performance of the local error indicators in driving adaptive mesh refinement algorithms based on different marking strategies. Finally, we analyze and test an inexpensive variant of the proposed error estimator which drastically reduces the overall computational cost of the adaptive procedures. Key words. Mimetic finite difference method, a posteriori error estimators, adaptive algorithms AMS subject classifications. 65N30, 65N15 1. Introduction. A
Approximation of nonlinear parabolic equations using a family of conformal and nonconformal schemes
, 2010
"... We consider a family of space discretisations for the approximation of nonlinear parabolic equations, such as the regularised mean curvature flow level set equation, using semiimplicit or fully implicit time schemes. The approximate solution provided by such a scheme is shown to converge thanks to ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We consider a family of space discretisations for the approximation of nonlinear parabolic equations, such as the regularised mean curvature flow level set equation, using semiimplicit or fully implicit time schemes. The approximate solution provided by such a scheme is shown to converge thanks to compactness and monotony arguments. Numerical examples show the accuracy of the method.
ReissnerMindlin
"... Calcolo manuscript No. (will be inserted by the editor) Numerical results for mimetic discretization of ..."
Abstract
 Add to MetaCart
Calcolo manuscript No. (will be inserted by the editor) Numerical results for mimetic discretization of