Guangri Xue

A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra

In this paper, we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell-centered finite difference scheme. A special non-symmetric quadrature rule is employed that yields a positive definite cell-centered system for the pressure by eliminating local velocities. The method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids, including hexahedra with non-planar faces. Theoretical and numerical results indicate first-order convergence for the pressure and face fluxes.

A Family of Multipoint Flux Mixed Finite Element Methods for Elliptic Problems on General Grids

Abstract In this paper, we discuss a family of multipoint flux mixed finite element (MFMFE) methods on simplicial, quadrilateral, hexahedral, and triangular-prismatic grids. The MFMFE methods are locally conservative with continuous normal fluxes, since they are developed within a variational framework as mixed finite element methods with special approximating spaces and quadrature rules. The latter allows for local flux elimination giving a cell-centered system for the scalar variable. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a non-symmetric quadrature rule on rough grids. Theoretical and numerical results demonstrate first order convergence for problems with full-tensor coeffcients. Second order superconvergence is observed on smooth grids.

Efficient algorithms for multiscale modeling in porous media

We describe multiscale mortar mixed finite element discretizations for second-order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. We discuss the construction of multiscale mortar basis and extend this concept to nonlinear interface operators. We present a multiscale preconditioning strategy to minimize the computational cost associated with construction of the multiscale mortar basis. We also discuss the use of appropriate quadrature rules and approximation spaces to reduce the saddle point system to a cell-centered pressure scheme. In particular, we focus on multiscale mortar multipoint flux approximation method for general hexahedral grids and full tensor permeabilities. Numerical results are presented to verify the accuracy and efficiency of these approaches. Copyright

Accurate Cell-Centered Discretizations for Modeling Multiphase Flow in Porous Media on General Hexahedral and Simplicial Grids

We introduce an accurate cell-centered method for modeling Darcy flow on general quadrilateral, hexahedral, and simplicial grids. We refer to these discretizations as the multipoint-flux mixed-finiteelement (MFMFE) method. The MFMFE method is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite-element method with special approximating spaces and quadrature rules. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a nonsymmetric quadrature rule on rough grids. The framework allows for handling hexahedral grids with nonplanar faces defined by trilinear mappings from the reference cube. Moreover, the MFMFE method allows for local elimination of the velocity, which leads to a cell-centered pressure system. Theoretical and numerical results demonstrate first-order convergence on rough grids. Second-order superconvergence is observed on smooth grids. We also discuss a new splitting scheme for modeling multiphase flows that can treat higher-order transport discretizations for saturations. We apply the MFMFE method to obtain physically consistent approximations to the velocity and a reference pressure on quadrilateral or hexahedral grids, and a discontinuous Galerkin method for saturations. For higher-order saturations, we propose an efficient post-processing technique that gives accurate velocities in the interior of the gridblocks. Computational results are provided for flow in highly heterogeneous reservoirs, including different capillary pressures arising from different rock types.

Benchmark 3D: A multipoint flux mixed finite element method on general hexahedra

In this paper we discuss a family of numerical schemes for modeling Darcy flow, the multipoint flux mixed finite element (MFMFE) methods. The MFMFE methods allow for an accurate and efficient treatment of irregular geometries and heterogeneities such as faults, layers, and pinchouts that require highly distorted grids and discontinuous coefficients. The methods can be reduced to cell-centered discretizations and have convergent pressures and velocities on general hexahedral and simplicial grids.

Preconditioning for Mixed Finite Element Formulations of Elliptic Problems

In this paper, we discuss a preconditioning technique for mixed finite element discretizations of elliptic equations. The technique is based on a block-diagonal approximation of the mass matrix which maintains the sparsity and positive definiteness of the corresponding Schur complement. This preconditioner arises from the multipoint flux mixed finite element method and is robust with respect to mesh size and is better conditioned for full permeability tensors than a preconditioner based on a diagonal approximation of the mass matrix.

Role of Computational Science in Protecting the Environment: Geological Storage of CO2

Simulation of field-scale CO2 sequestration (which is defined as the capture, separation and long-term storage of CO2 for environmental purposes) has gained significant importance in recent times. Here we discuss mathematical and computational formulations for describing reservoir characterization and evaluation of long term CO2 storage in saline aquifers as well as current computational capabilities and challenges.