Numerical Simulation of Two-Phase Flow through Heterogeneous Porous Media

A mixed finite element method is combined to finite volume schemes on structured and unstructured grids for the approximation of the solution of incompressible flow in heterogeneous porous media. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a nonlinear degenerate diffusion–convection equation arising in modeling of flow and transport in porous media.     

A mixed hybrid formulation based on oscillated finite element polynomials for solving Helmholtz problems

A mixed-hybrid-type formulation is proposed for solving Helmholtz problems. This method is based on (a) a local approximation of the solution by oscillated finite element polynomials and (b) the use of Lagrange multipliers to “weakly” enforce the continuity across element boundaries. The computational complexity of the proposed discretization method is determined mainly by the total number of Lagrange multiplier degrees of freedom introduced at the interior edges of the finite element mesh, and the sparsity pattern of the corresponding system matrix. Preliminary numerical results are reported to illustrate the potential of the proposed solution methodology for solving efficiently Helmholtz problems in the mid- and high-frequency regimes.     

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives the solution in an efficient and unconditionally stable way.     

A posteriori error estimation and adaptive computation of conduction convection problems

In this paper, an adaptive finite element method is developed for stationary conduction convection problems. Using a mixed finite element formulation, residual type a posteriori error estimates are derived by means of the general framework of R. Verfürth. The effectiveness of the adaptive method is further demonstrated through two numerical examples. The first example is problem with known solution and the second example is a physical model of square cavity stationary flow.     

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.     

On exact and approximate boundary controllabilities for the heat equation: A numerical approach

The present article is concerned with the numerical implementation of the Hilbert uniqueness method for solving exact and approximate boundary controllability problems for the heat equation. Using convex duality, we reduce the solution of the boundary control problems to the solution of identification problems for the initial data of an adjoint heat equation. To solve these identification problems, we use a combination of finite difference methods for the time discretization, finite element methods for the space discretization, and of conjugate gradient and operator splitting methods for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of exact and approximate boundary controllability test problems in two space dimensions. The numerical results validate the methods discussed in this article and clearly show the computational advantage of using second-order accurate time discretization methods to approximate the control problems.     

The influence of integration rule accuracy on the calculation of surface subsidence by the nucleus of strain method in conjunction with a finite element reservoir simulator

Relative accuracy of numerical quadrature rules when applied to the simulation of underground petroleum reservoirs by means of the finite element method is investigated. Fluid flow within the reservoir is calculated via the finite element method and the resulting deformation by the nucleus of strain technique. By analysing a simple problem it was found that the solution method was susceptible to changes in numerical quadrature for reservoirs that were positioned near the ground surface and that care is required when solving such problems due to the singularities occurring in the integrands which appear in the nucleus of strain approach.     

二阶椭圆问题的一类广义有限元法

本文提出了求解二阶椭圆问题的一类广义有限元方法,分析了广义有限元方法的优越性,证明了二阶椭圆问题的广义有限元方法具有比标准的Galerkin有限元方法更高阶的收敛速度,根据插值算子的性质,进一步证明了有限元解的亏量迭代校正收敛到广义有限元解,并用数值例子说明广义有限元方法是有效的.     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.     

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L ² and the H ¹ norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.     

Well-posedness and finite element approximation of time dependent generalized bioconvective flow

We consider the finite element approximation of a time dependent generalized bioconvective flow. The underlying system of partial differential equations consists of incompressible Navier–Stokes type convection equations coupled with an equation describing the transport of micro-organisms. The viscosity of the fluid is assumed to be a function of the concentration of the micro-organisms. We show the existence and uniqueness of the weak solution of the system in two dimensions and construct numerical approximations based on the finite element method, for which we obtain error estimates. In addition, we conduct several numerical experiments to demonstrate the accuracy of the numerical method and perform simulations of the bioconvection pattern formations based on realistic model parameters to demonstrate the validity of the proposed numerical algorithm.     

基于非结构自适应网格的复合有限体积法

利用文献[1]中将Lax-Wendroff格式和Lax-Friedrichs格式整体复合作用构成二维无结构网格上的复合型有限体积法,同时利用Delaunay方法,根据流场流动特性变化的梯度值为指示器对网格进行加密和粗化,实现自适应,并将此方法应用到二维浅水波方程的求解上,进行了二维部分溃坝,倾斜水跃的数值实验.结果表明,该方法是一个计算稳定、能适应复杂的求解域、能很好地捕捉激波、且计算速度快的算法.     

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g.around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two-dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl-Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

Leak identification in a saturated unsteady flow via a Cauchy problem

In this initial study, we propose a numerical method for identifying multiple leak zones in a saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modeled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes because the leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we modify it and employ a previously proposed iterative regularizing method. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint to obtain a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a finite element method and the numerical results indicate that the leak zones can be identified with the proposed method.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Two-level Schwarz method for unilateral variational inequalities

The numerical solution of variational inequalities of obstacletype associated with second-order elliptic operators is considered.Iterative methods based on the domain decomposition approachare proposed for discrete obstacle problems arising from thecontinuous, piecewise linear finite element approximation ofthe differential problem. A new variant of the Schwarz methodology,called the two-level Schwarz method, is developed offering thepossibility of making use of fast linear solvers (e.g., linearmultigrid and fictitious domain methods) for the genuinely nonlinearobstacle problems. Namely, by using particular monotonicityresults, the computational domain can be partitioned into (mesh)subdomains with linear and nonlinear (obstacle-type) subproblems.By taking advantage of this domain decomposition and fast linearsolvers, efficient implementation algorithms for large-scalediscrete obstacle problems can be developed. The last part ofthe paper is devoted to illustrate numerical experiments.     

Free vibration and buckling analysis of plates using B-spline wavelet on the interval Mindlin element

A finite element method (FEM) of B-spline wavelet on the interval (BSWI) is used in this paper to solve the free vibration and buckling problems of plates based on Reissner–Mindlin theory. By aid of the high accuracy of B-spline functions approximation for structural analysis, the proposed method could obtain a fast convergence and a satisfying numerical accuracy with fewer degrees of freedoms (DOF). The numerical examples demonstrate that the present BSWI method achieves the high accuracy compared to the exact solution and others existing approaches in the literatures. The BSWI finite element has potential to be used as a numerical method in analysis and design.     

Constructing Order Two Superconvergent WG Finite Elements on Rectangular Meshes

In this paper, we introduce a stabilizer free weak Galerkin (SFWG) finite element method for second order elliptic problems on rectangular meshes. With a special weak Gradient space, an order two superconvergence for the SFWG finite element solution is obtained, in both $L^2$ and $H^1$ norms. A local post-process lifts such a $P_k$ weak Galerkin solution to an optimal order $P_{k+2}$ solution. The numerical results confirm the theory.     

Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.