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A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium by employing a mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics (CEMFEM) for the concentration equation. The key point is to use a two-grid scheme to linearize the nonlinear term in the coupling equations. The main procedure of the algorithm is to solve small scaled nonlinear equations on the coarse grid and to deal with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Then, it is shown that a two-grid algorithm achieves optimal approximation as long as the mesh sizes satisfy H = O ( h 1 2 ) . Finally, numerical experiments confirmed the numerical analysis of two-grid algorithm.

In this article, we propose an expanded mixed finite element method for variable-coefficient fractional dispersion equations (FDEs). By introducing two intermediate variables, p=Du and σ=−Iθβp, the FDEs are reformulated into a mixed system involving only lower-order derivatives. Based on this, we construct an expanded mixed variational framework and prove the weak coercivity in the sense of the LBB condition over appropriately chosen Sobolev spaces, thereby ensuring the well-posedness of the formulation. Then, we develop an expanded mixed finite element scheme and prove that the unique expanded finite element solution possesses optimal approximation accuracy to the fractional flux σ, the gradient p and the unknown u. Finally, numerical experiments are conducted to verify the efficiency and accuracy of the proposed method.

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size and the fine grid size satisfy ( ), where is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

In this paper, a second-order hyperbolic equation is solved by a two-grid algorithm combined with the expanded mixed finite element method. The error estimate of the expanded mixed finite element method with discrete-time scheme is demonstrated. Moreover, we present a two-grid method and analyze its convergence. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h12). Finally, some numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method.

A new extended mixed finite element method (MFEM) is suggested for parabolic integrodifferential equations (PIDEs) with nonlinear memory. On the contrary, of the extended mixed scheme, the modern extended mixed element system refers to an asymmetric positive well known and the two gradient equation as well as the flux equation can be isolated from the scalar undefined equation. The presence and uniqueness of the semi-discrete system can be confirmed and error estimates can be achieved for semidiscrete. Fully discrete can be said to be discretization.

This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control-volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods. [PUBLICATION ABSTRACT]

We develop a family of expanded mixed multiscale finite element methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed multiscale finite element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity, and Lagrange multipliers. We use multiscale basis functions for both the velocity and the gradient of pressure. In the expanded mixed MsFEM framework, we consider both separable and nonseparable spatial scales. Specifically, we analyze the methods in three categories: periodic separable scales, G-convergent separable scales, and a continuum of scales. When there is no scale separation, using some global information can significantly improve the accuracy of the expanded mixed MsFEMs. We present a rigorous convergence analysis of these methods that includes both conforming and nonconforming formulations. Numerical results are presented for various multiscale models of flow in porous media with shale barriers that illustrate the efficacy of the proposed family of expanded mixed MsFEMs. [PUBLICATION ABSTRACT]

In this paper, we propose an Expanded Characteristic-mixed Finite Element Method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection part in time and an expanded mixed finite element spatial approximation to deal with the diffusion part. The scheme is stable since fluid is transported along the approximate characteristics on the discrete level. At the same time it expands the standard mixed finite element method in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux. Our analysis shows the method approximates the scalar unknown, its gradient, and its flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. A numerical example is presented to show that the scheme is of high performance.

A new expanded mixed finite element method is constructed for solving the Kirchhoff type parabolic equation. Compared with the traditional mixed element methods, this method can deal well with the case that the diffusion coefficient is relatively small, and the coefficient matrix is symmetric positive definite. The regularity of the solution of the Kirchhoff type parabolic equation is considered, and the stability of the semidiscrete scheme is analyzed. Based on backward Euler difference procedure in time, a fully discrete algorithm is provided, and then the existence and uniqueness of the solution are proved by Brouwer’s fixed point theorem. Meanwhile, the optimal error estimates in L 2 -norm and H (div)-norm are derived. Finally, some numerical examples are presented to verify the effectiveness of the proposed algorithms with two-grid technique.

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.