An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L²$L^2$-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.     

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media. The flow equation is solved by the mixed finite element method, and the transport equation is approximated by a discontinuous Galerkin method. The scheme is continuous in time and a priori hp error estimates is presented.     

Analysis of a splitting method for incompressible inviscid rotational flow problems

This paper is devoted to analyze a splitting method for solving incompressible inviscid rotational flows. The problem is first recast into the velocity–vorticity–pressure formulation by introducing the additional vorticity variable, and then split into three consecutive subsystems. For each subsystem, the L²$L^2$ least-squares finite element approach is applied to attain accurate numerical solutions. We show that for each time step this splitting least-squares approach exhibits an optimal rate of convergence in the H¹$H^1$ norm for velocity and pressure, and a suboptimal rate in the L²$L^2$ norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates.     

A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations

Summary. We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious domain and the boundary mesh are independent, up to a mesh-length ratio. Received February 24, 1999 / Revised version received January 30, 2000 / Published online October 16, 2000     

A mortar mimetic finite difference method on non-matching grids

We consider mimetic finite difference approximations to second order elliptic problems on non-matching multiblock grids. Mortar finite elements are employed on the non-matching interfaces to impose weak flux continuity. Optimal convergence and, in certain cases, superconvergence is established for both the scalar variable and its flux. The theory is confirmed by computational results. Supported by the US Department of Energy, under contractW-7405-ENG-36. LA-UR-04-4740. Partially supported by NSF grants EIA 0121523 and DMS 0411413, by NPACI grant UCSD 10181410, and by DOE grant DE-FGO2-04ER25617. Partially supported by NSF grants DMS 0107389, DMS 0112239 and DMS 0411694 and by DOE grant DE-FG02-04ER25618.     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Finite element method for incompressible miscible displacement in porous media

Under the assumptions of nonlinear finite element and t =o(h), Ewing and Wheeler discussed a Galerkin method for the single phase incompressible miscible displacement of one fluid by another in porous media. In the present paper we give a finite element scheme which weakens the t =o(h)-restriction to t =o(H ), 0 < 1/2. Furthermore, this scheme is suitable for both linear element and nonlinear element. We also derive the optimal approximation estimates for concentrationc, its gradient c and the gradient p of the pressurep.     

Discontinuous Least-Squares finite element method for the Div-Curl problem

In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method.     

Error estimates for a mixed finite volume method for the p-Laplacian problem

In this work we propose and analyze a mixed finite volume method for the p-Laplacian problem which is based on the lowest order Raviart–Thomas element for the vector variable and the P1 nonconforming element for the scalar variable. It is shown that this method can be reduced to a P1 nonconforming finite element method for the scalar variable only. One can then recover the vector approximation from the computed scalar approximation in a virtually cost-free manner. Optimal a priori error estimates are proved for both approximations by the quasi-norm techniques. We also derive an implicit error estimator of Bank–Weiser type which is based on the local Neumann problems.This work was supported by the Post-doctoral Fellowship Program of Korea Science & Engineering Foundation (KOSEF).     

Two-grid finite volume element method for linear and nonlinear elliptic problems

Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms. A set of numerical examples are presented to confirm the estimates. The work is supported by the National Natural Science Foundation of China (Grant No: 10601045).     

Superconvergence of mixed covolume method for elliptic problems on triangular grids

In this paper, we consider the superconvergence of a mixed covolume method on the quasi-uniform triangular grids for the variable coefficient-matrix Poisson equations. The superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained. With these superconvergence estimates, we establish the superconvergence estimates and the L^∞$L^{\infty }$-error estimates for the mixed covolume method for the elliptic problems. Based on the superconvergence of the mixed covolume method, under the condition that the triangulation is uniform, we construct a post-processing method for the approximate velocity which improves the order of approximation of the approximate velocity.     

A discontinuous Galerkin method on refined meshes for the two-dimensional time-harmonic Maxwell equations in composite materials

In this paper, a discontinuous Galerkin method for the two-dimensional time-harmonic Maxwell equations in composite materials is presented. The divergence constraint is taken into account by a regularized variational formulation and the tangential and normal jumps of the discrete solution at the element interfaces are penalized. Due to an appropriate mesh refinement near exterior and interior corners, the singular behaviour of the electromagnetic field is taken into account. Optimal error estimates in a discrete energy norm and in the L²$L^2$-norm are proved in the case where the exact solution is singular.     

A mixed finite element method for solving the nonstationary stokes equation

Summary This paper deals with a mixed finite element method for approximating a fourth order initial value problem arising from the nonstationary Stokes problem. For piecewise linear shape functions error estimates are given with convergence rates similar to the elliptic case. Some numerical computations will illustrate the theoretical results.     

A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems

We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes.     

Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation

The classical Hu–Washizu mixed formulation for plane problems in elasticity is examined afresh, with the emphasis on behavior in the incompressible limit. The classical continuous problem is embedded in a family of Hu–Washizu problems parametrized by a scalar α for which corresponds to the classical formulation, with λ and μ being the Lamé parameters. Uniform well- posedness in the incompressible limit of the continuous problem is established for α ≠ − 1. Finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes. Conditions for uniform convergence are made explicit. These conditions are shown to be met by particular choices of bases for stresses and strains, and include bases that are well known, as well as newly constructed bases. Though a discrete version of the spherical part of the stress exhibits checkerboard modes, it is shown that a λ-independent a priori error estimate for the displacement can be established. Furthermore, a λ-independent estimate is established for the post-processed stress. The theoretical results are explored further through selected numerical examples.     

A second order finite difference error indicator for adaptive transonic flow computations

Summary. An adaptive finite element method for the calculation of transonic potential flows was developed. An error indicator based on first order finite differences of gradients is introduced as a local error estimator. It measures second order distributional derivatives. Estimates involving this error estimator, a residual and the error are given. The error estimator can be used as a criterion for mesh refinement. We also give some computational results. Received September 16, 1993 / Revised version received June 7, 1994     

Error analysis of an enhanced DtN-FE method for exterior scattering problems

In this work we analyze the convergence of the high-order Enhanced DtN-FEM algorithm, described in our previous work (Nicholls and Nigam, J. Comput. Phys. 194:278–303, 2004), for solving exterior acoustic scattering problems in . This algorithm consists of using an exact Dirichlet-to-Neumann (DtN) map on a hypersurface enclosing the scatterer, where the hypersurface is a perturbation of a circle, and, in practice, the perturbation can be very large. Our theoretical work had shown the DtN map was analytic as a function of this perturbation. In the present work, we carefully analyze the error introduced by virtue of using this algorithm. Specifically, we give a full account of the error introduced by truncating the DtN map at a finite order in the perturbation expansion, and study the well-posedness of the associated formulation. During computation, the Fourier series of the Dirichlet data on the artificial boundary must be truncated. To deal with the ensuing loss of uniqueness of solutions, we propose a modified DtN map, and prove well-posedness of the resulting problem. We quantify the spectral error introduced due to this truncation of the data. The key tools in the analysis include a new theorem on the analyticity of the DtN map in a suitable Sobolev space, and another on the perturbation of non-self-adjoint Fredholm operators.     

An adaptive discontinuous finite volume method for elliptic problems

An adaptive discontinuous finite volume method is developed and analyzed in this paper. We prove that the adaptive procedure achieves guaranteed error reduction in a mesh-dependent energy norm and has a linear convergence rate. Numerical results are also presented to illustrate the theoretical analysis.