Least-squares mixed finite element method for a class of stokes equation

ＩｎｔｒｏｄｕｃｔｉｏｎＡｇｅｎｅｒａｌｔｈｅｏｒｙｏｆｔｈｅｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｈａｓｂｅｅｎｄｅｖｅｌｏｐｅｄｂｙＡＫＡｚｉｚ,ＲＢＫｅｌｌｏｇｇａｎｄＡＢＳｔｅｐｈｅｎｓｉｎ[1].Ｔｈｅｍｏｓｔｉｍｐｏｒｔａｎｔａｄｖａｎｔａｇｅｌｅａｄｓｔｏａｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅｐｒｏｂｌｅｍ.ＪＨＢｒａｍｂｌｅａｎｄＪＡＮｉｔｓｈｅｐｒｅｓｅｎｔｅｄａｌｅａｓｔ＿ｓｑｕａｒｅｓｍｅｔｈｏｄｆｏｒＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｓｉｎ[2].Ｔｈｅｍｅｔｈｏｄｇｅ…     

A coupling method of difference with high order accuracy and boundary integral equation for evolutionary equation and its error estimates

In the present paper,a new numerical method for solving initial-boundary valueproblems of evolutionary equations is proposed and studied,combining difference methodwith high accuracy with boundary integral equation method.The numerical approximateschemes for both problems on a bounded or unbounded domain in R~3 are proposed and theirprior error estimates are obtained.     

The boundary layer method for the solution of singular perturbation problem for the parabolic partial differential equation

In this paper, we discuss the singular perturbation problem of the parabolic partial differential equation. As usual, we must reduce the mesh size in the neighbourhood of the boundary layer so that typical feature of the boundary layer will not be lost. Then we need very large operational quantity when mesh sizes are getting too small. Now we propose the boundary layer scheme, which need not take very fine mesh size in the neighbourhood of the boundary layer. Numerical examples show that the accuracy can be satisfied with moderate step size.     

Positive‐definite q‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

A new family of locally conservative cell‐centred flux‐continuous schemes is presented for solving the porous media general‐tensor pressure equation. A general geometry‐permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control‐volume distributed subcell flux‐continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2 :259–290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive‐definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical‐space schemes are shown to be non‐symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non‐symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical‐space and subcell‐space q‐families of schemes. M‐matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical‐space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell‐wise constant tensor schemes and that subcell tensor approximation using the control‐volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd.     

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

A family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2 :259–290; Comput. Geo. 2002; 6 :433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51 :1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007). In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve:

• (i) Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice.
• (ii) Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution.
• (iii) Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution.

The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.     

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

This paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least‐squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax–Milgram theorem, and besides proving the well‐posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart–Thomas elements of any order for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual‐based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle‐bubble and edge‐bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators and showing the behavior of the associated adaptive algorithms, are reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Difference scheme and numerical simulation based on mixed finite element method for natural convection problem

ntroductionLetΩ R2 beaboundeddomain .Weconsiderthefollowingnon_stationarynaturalconvectionproblem :Problem (Ⅰ )　Findu =(u1,u2 ) ,p ,andTsuchthat,foranyt1>0 ,ut- μΔu +(u· )u + p=λjT　　 ((x ,y ,t) ∈Ω× (0 ,t1) ) ,divu =0　　　　　　　　　 ((x ,y,t) ∈Ω× (0 ,t1) ) ,Tt-ΔT +λu· T =0　　 ((x,y,t) ∈Ω× (0 ,t1) ) ,u =0 ,T =0　　　　　　 ((x,y,t)∈ Ω× (0 ,t1) ) ,u(x ,y ,0 ) =0 ,　T(x,y,0 ) =f(x,y)　　 ((x,y) ∈Ω) ,whereuisthefluidvelocityvectorfield ,pthepressurefield ,Tthet…     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION （Ⅰ）——THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.     

Linearized and rational approximation method for solving non‐linear Burgers' equation

A new numerical method called linearized and rational approximation method is presented to solve non‐linear evolution equations. The utility of the method is demonstrated for the case of differentiation of functions involving steep gradients. The solution of Burgers' equation is presented to illustrate the effectiveness of the technique for the solution of non‐linear evolution equations exhibiting nearly discontinuous solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Residual a posteriori error estimate two-grid methods for the steady Navier-Stokes equation with stream function form

Residual based on a posteriori error estimates for conforming element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit characteristic‐based finite volume element method for level set equation

Accurate modeling of interfacial flows requires a realistic representation of interface topology. To reduce the computational effort from the complexity of the interface topological changes, the level set method is widely used for solving two‐phase flow problems. This paper presents an explicit characteristic‐based finite volume element method for solving the two‐dimensional level set equation. The method is applicable for the case of non‐divergence‐free velocity field. Accuracy and performance of the proposed method are evaluated via test cases with prescribed velocity fields on structured grids. By given a velocity field, the motion of interface in the normal direction and the mean curvature, examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time. Copyright © 2010 John Wiley & Sons, Ltd.     

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method (FEM) based on a proper orthogonal decomposition (POD) theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save memory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element (FE) scheme is shown to be unconditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations (FDEs).     

Numerical solution of a nonlinear reaction-diffusion equation

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkinfinite element method,which has been proved to be2nd-order accurate in time and4th-orderin space.The comparison between the exact and numerical solutions of progressive wavesshows that this numerical scheme is quite accurate,stable and efficient.It is also shown thatany local disturbance will spread,have a full growth and finally form two progressive wavespropagating in both directions.The shape and the speed of the long term progressive wavesare determined by the system itself,and do not depend on the details of the initial values.     

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.     

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition.A linearized map in a closed convex set is defined.The image set is precompact,and thus a fixed point exists.A multi-scale expansion method is used to obtain the homogenized equation.This equation satisfies a similar growth condition.     

General digital picture processing and texture analysis for photoelasticity

In this paper, a general digital picture processing and texture analysis for photoelasticity is developed. Overcoming some defects of references [2] and [3], it presents an effective method to analyse fringe patterns of the photoelasticity. By means of the trigonometric function relationship between the light intensity I and the image fringe order N, the equations of the fringe order N on brightness Z are deduced, and the mechanical parameters are thus obtained. We established a system of sigital picture processing and texture analysis for photoelasticity, which is called OYC-1 system. Finally, this system is checked with an example. It is found that the differences between measured results and the theoretical values are within 2.3 percent.This paper is supported by the science funds of the Academia Sinica.     

Highly efficient H1-Galerkin mixed finite element method （MFEM） for parabolic integro-differential equation

A highly efficient H1-Galerkin mixed finite element method(MFEM) is presented with linear triangular element for the parabolic integro-differential equation.Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O(h2) for both the original variable u in H1(π) norm and the flux p =u in H(div,π) norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O(h3) are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.     

A NONLINEAR GALERKIN MIXED ELEMENT METHOD AND A POSTERIORI ERROR ESTIMATOR FOR THE STATIONARY NAVIER-STOKES EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｎｏｎｌｉｎｅａｒＧａｌｅｒｋｉｎｍｅｔｈｏｄｉｓａｍｕｌｔｉ＿ｌｅｖｅｌｓｃｈｅｍｅｔｏｆｉｎｄｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｆｏｒｔｈｅｄｉｓｓｉｐａｔｉｖｅＰＤＥ (ｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ) .Ｔｈｉｓｍｅｔｈｏｄｃｏｎｓｉｓｔｓｉｎｓｐｌｉｔｔｉｎｇｔｈｅｕｎｋｎｏｗｎｉｎｔｏｔｗｏ (ｏｒｍｏｒｅ)ｔｅｒｍｓ ,ｗｈｉｃｈｂｅｌｏｎｇｔｏｔｈｅｄｉｓｃｒｅｔｅｓｐａｃｅｓｗｉｔｈｄｉｆｆｅｒｅｎｔｍｅｓｈｓｉｚｅ .Ｔｈｅ…     

Criteria for finite element algorithm of generalized heat conduction equation

To eliminate oscillation and overbounding of finite element solutions of classical heat conduction equation, the author and Xiao have put forward two new concepts of monotonies and have derived and proved several criteria. This idea is borrowed here to deal with generalized heat conduction equation and finite element criteria for eliminating oscillation and overbounding are also presented. Some new and useful conclusions are drawn.