Multipoint Flux Mixed Finite Element Research Articles

An online generalized multiscale approximation of the multipoint flux mixed finite element method

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method.

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

A multipoint flux mixed finite element method with mass-conservative characteristic finite element method for incompressible miscible displacement problem

Multipoint flux mixed finite element method for parabolic optimal control problems

<abstract><p>In this paper, we research semi-discrete multipoint flux mixed finite element (MFMFE) method for parabolic optimal control problem (OCP). The state and co-state variables are approximated by the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element (MFE) spaces and the control is approximated by piecewise constant. The advantage of this type of mixed element method is that it can decouple the state and adjoint state variables as cell-centered difference schemes rather than to solve saddle point algebraic equations. Error estimates and convergence orders are derived rigorously for state and control variables. Finally, a numerical example is given to confirm our theoretical analysis.</p></abstract>

Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT12) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a multiscale method for the Darcy–Forchheimer model in highly heterogeneous porous media. The problem is solved in the framework of generalized multiscale finite element method (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. We consider a MFMFE method that utilizes the lowest order Brezzi–Douglas–Marini (BDM1) mixed finite element spaces for the approximation of velocity and pressure. The symmetric trapezoidal quadrature rule is employed for the integral of bilinear forms related to velocity variables so that the local velocity elimination is allowed which leads to a cell-centered system for pressure. We construct the multiscale space for pressure and solve the problem on the coarse grid following the GMsFEM framework. In the offline stage, we construct local snapshot spaces and perform spectral decompositions to get the offline space with a smaller dimension. In the online stage, we use Newton iterative algorithm to solve the nonlinear problem and obtain the offline solution, which reduces the number of iterations greatly compared to the standard Picard iterative algorithm. Based on the offline basis functions and the offline solution, we calculate online basis functions on each coarse element to enrich the multiscale space iteratively. The online basis functions contain the important global information and are effective to reduce relative errors substantially. Numerical examples are provided to highlight the performance of the proposed multiscale method.

A coupled multipoint stress–multipoint flux mixed finite element method for the Biot system of poroelasticity

We present a mixed finite element method for a five-field formulation of the Biot system of poroelasticity that reduces to a cell-centered pressure–displacement system on simplicial and quadrilateral grids. A mixed stress–displacement–rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity–pressure Darcy formulation. The spatial discretization is based on combining the multipoint stress mixed finite element (MSMFE) method for elasticity and the multipoint flux mixed finite element (MFMFE) method for Darcy flow. It uses the lowest order Brezzi–Douglas–Marini mixed finite element spaces for the poroelastic stress and Darcy velocity, piecewise constant displacement and pressure, and continuous piecewise linear or bilinear rotation. A vertex quadrature rule is applied to the velocity, stress, and stress–rotation bilinear forms, which block-diagonalizes the corresponding matrices and allows for local velocity, stress, and rotation elimination. This leads to a cell-centered positive-definite system for pressure and displacement at each time step. We perform error analysis for the semidiscrete and fully discrete formulations, establishing first order convergence for all variables in their natural norms. The numerical tests confirm the theoretical convergence rates and illustrate the locking-free property of the method.

A Multipoint Stress Mixed Finite Element Method for Elasticity on Simplicial Grids

We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element ...

A Multipoint Flux Mixed Finite Element Method for Darcy–Forchheimer Incompressible Miscible Displacement Problem

We consider a numerical scheme for incompressible miscible displacement problem in porous media. A multipoint flux mixed finite element method is used to handle the velocity–pressure equation. The standard finite element method is used to approximate the concentration equation. Error estimates for pressure and velocity and concentration are presented. Numerical experiments show that the convergence rates of this scheme are in agreement with the theoretical analysis.

Comparison of the hybrid and multipoint‐flux method

AbstractWe compare two approaches for the efficient implementation of the mixed finite element method with applications for elliptic problems, like single phase flow, and hyperbolic problems, like the wave equations. We briefly explain the hybrid and the multipoint flux mixed finite element method (MFMFE) and conduct a comparison of the two methods in terms of efficiency.

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L2 projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.

Multipoint flux mixed finite element methods for slightly compressible flow in porous media

In this paper, we consider multipoint flux mixed finite element discretizations for slightly compressible Darcy flow in porous media. The methods are formulated on general meshes composed of triangles, quadrilaterals, tetrahedra or hexahedra. An inexact Newton method that allows for local velocity elimination is proposed for the solution of the nonlinear fully discrete scheme. We derive optimal error estimates for both the scalar and vector unknowns in the semidiscrete formulation. Numerical examples illustrate the convergence behavior of the methods, and their performance on test problems including permeability coefficients with increasing heterogeneity.

Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

We perform a convergence analysis of a two-grid fixed stress split algorithm for the Biot system modeling coupled flow and deformation in heterogeneous poroelastic media. The two-grid fixed stress split scheme solves the flow subproblem on a fine grid using a multipoint flux mixed finite element method by imposing the fixed mean stress constraint followed by the poromechanics subproblem on a coarse grid using a conforming Galerkin method in every coupling iteration at each time step. Restriction operators map the fine scale flow solution to the coarse scale poromechanical grid and prolongation operators map the coarse scale poromechanical solution to the fine scale flow grid. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the two-grid scheme is a contraction map under certain conditions. Those conditions are used to construct the restriction and prolongation operators as well as arrive at effective elastic properties for the coarse scale poromechanical solve in terms of the fine scale elastic properties. We analyze the contraction map with a numerical result comparing the numerically computed to the theoretically obtained contraction constant for a poroelastic medium with substantial spatial variability in the poroelastic moduli.

Convergence analysis of fixed stress split iterative scheme for anisotropic poroelasticity with tensor Biot parameter

We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy.

A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs

In coupled flow and poromechanics phenomena representing hydrocarbon production or CO2 sequestration in deep subsurface reservoirs, the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs. The typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock. The former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs. In order to address these challenges, we augment the fixed-stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids. Flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming Galerkin method. The multiscale operators are constructed using a procedure that involves singular value decompositions, a surface intersections algorithm and Delaunay triangulations. We numerically demonstrate the convergence of the augmented scheme using the classical Mandel's problem solution.

A multipoint flux mixed finite element method for the compressible Darcy–Forchheimer models

We develop a multipoint flux mixed finite element method for the compressible Darcy–Forchheimer models. This method is motivated by the multipoint flux approximation method. Based on the lowest order Brezzi–Douglas–Marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to a cell-centered system for the pressures. We develop two approximation methods for the problems, the semi-discrete method and fully discrete method, where the fully discrete method includes the backward Euler scheme and the Crank–Nicolson scheme. Theoretical results show the convergences for both pressure and velocity. The numerical experiments show that the convergence rates of the method are in agreement with the theoretical analysis.

A parallel framework for a multipoint flux mixed finite element equation of state compositional flow simulator

Mathematical models of physical problems are becoming increasingly complex and computationally intensive. At the same time, computing hardware is becoming more parallelized with an increasing number of cores promoting simultaneous tasks. In this work, we present a parallel, equation of state (EOS), compositional flow simulator for evaluating CO2 sequestration, enhanced oil recovery techniques such as gas flooding, and other subsurface porous media applications. Using the multipoint flux mixed finite element (MFMFE) method for spatial discretization, it can handle complex reservoir geometries using general distorted hexahedral grid elements, as well as satisfy local mass conservation and compute accurate phase fluxes. A parallel framework for the MFMFE is presented that has been extended to the highly non-linear, EOS, compositional flow model. Much of the non-linearity is due to the local flash and stability calculations associated with interphase mass transfer and phase behavior. Parallel multigrid linear solver libraries such as HYPRE are utilized to solve the algebraic problems on each Newton step. We perform a variety of strong and weak parallel scaling studies up to 10 million elements and 1024 processors, and discuss possible load balancing issues.

A posteriori error analysis of multipoint flux mixed finite element methods for interface problems

In this paper, the multipoint flux mixed finite element method is used to approximate the flux of two-dimensional elliptic interface problems. Within the class of modified quasi-monotonically distributed coefficients, we derive uniformly robust residual-type a posteriori error estimators for the flux error. Based on the residual-type estimator, we further develop robust implicit and explicit recovery-type estimators through gradient recovery in H(curl) conforming finite element spaces. Numerical experiments are presented to support the theoretical results.

Residual-based a posteriori error estimation for multipoint flux mixed finite element methods

A novel residual-type {\it a posteriori} error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions. The derived {\it a posteriori} error estimator for the velocity and pressure error in $L^{2}-$norm consists of discretization and quadrature indicators, and is shown to be reliable and efficient. The main tools of analysis are a locally postprocessed approximation to the pressure solution of an auxiliary problem and a quadrature error estimate. Numerical experiments are presented to illustrate the competitive behavior of the estimator.

Compositional flow modeling using a multi-point flux mixed finite element method

We present a general compositional formulation using multi-point flux mixed finite element (MFMFE) method on general hexahedral grids. The mixed finite element framework allows for local mass conservation, accurate flux approximation, and a more general treatment of boundary conditions. The multi-point flux inherent in MFMFE scheme allows the usage of a full permeability tensor. The proposed formulation is an extension of single and two-phase flow formulations presented by Wheeler and Yotov, SIAM J. Numer. Anal. 44(5), 2082–2106 (35) with similar convergence properties. Furthermore, the formulation allows for black oil, single-phase and multi-phase incompressible, slightly and fully compressible flow models utilizing the same design for different fluid systems. An accurate treatment of diffusive/dispersive fluxes owing to additional velocity degrees of freedom is also presented. The applications areas of interest include gas flooding, CO 2 sequestration, contaminant removal, and groundwater remediation.