A STABILIZED CRANK-NICOLSON MIXED FINITE VOLUME ELEMENT FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER-STOKES EQUATIONS

A time semi-discrete Crank-Nicolson （CN） formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element （SCNMFVE） formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.

A STABILIZED MIXED FINITE ELEMENT FORMULATION FOR THE NON-STATIONARY INCOMPRESSIBLE BOUSSINESQ EQUATIONS

In this study, we employ mixed finite element （MFE） method, two local Gauss integrals, and parameter-free to establish a stabilized MFE formulation for the non-stationary incompressible Boussinesq equations. We also provide the theoretical analysis of the existence, uniqueness, stability, and convergence of the stabilized MFE solutions for the stabilized MFE formulation.

Discrete formulation of mixed finite element methods for vapor deposition chemical reaction equations

The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.

THEORETICAL BASIS AND GENERAL OPTIMAL FORMULATIONS OF ISOPARAMETRIC GENERALIZED HYBRID/MIXED ELEMENT MODEL FOR IMPROVED STRESS ANALYSIS

By the modified three-field Hu-Washizu principle, this paper establishes a theoretical founda- tion and general convenient formulations to generate convergent stable generalized hybrid/mixed cle- ment (GH/ME) model which is invariant with respect to coordinate, insensitive to geometric distortion and suitable for improved stress computation. In the two proposed formulations, the stress equilibrium and orthogonality constraints are imposed through incompatible displacement and internal strain modes respectively. The proposed model by the general formulations in this paper is characterized by including as- sumed stress/strain, assumed stress, variable-node, singular, compatible and incompatible GH/ME models. When using regular meshes or the constant values of the isoparametric Jacobian Det in the assumed strain in- terpolation, the incompatible GH/ME model degenerates to the hybrid/mixed element model. Both general and concrete guidelines for the optimal selection of element shape functions are suggested. By means of the GH/ME theory in this paper, a family of new GH/ME can be and have been easily constructed. The software can also be developed conveniently because all the standard subroutines for the corresponding isoparametric displacement elements can be utilized directly.

ANALYSIS OF AUGMENTED THREE-FIELD MACRO-HYBRID MIXED FINITE ELEMENT SCHEMES

On the basis of composition duality principles, augmented three-field macrohybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladysenskaja-Babulka-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decompositions. Then, through compositional dualization macro-hybrid mixed problems are obtained, with internal boundary dual traces as Lagrange multipliers. Also, ＂mass＂ preconditioned aug- mentation of three-field formulations are derived, stabilizing macro-hybrid mixed finite element schemes and rendering possible speed up of rates of convergence. Dual mixed incompressible Darcy flow problems illustrate the theory throughout the paper.

Multiscale Hybrid-Mixed Finite Element Method for Flow Simulation in Fractured Porous Media

The multiscale hybrid-mixed(MHM)method is applied to the numerical approximation of two-dimensional matrix fluid flow in porous media with fractures.The two-dimensional fluid flow in the reservoir and the one-dimensional flow in the discrete fractures are approximated using mixed finite elements.The coupling of the two-dimensional matrix flow with the one-dimensional fracture flow is enforced using the pressure of the one-dimensional flow as a Lagrange multiplier to express the conservation of fluid transfer between the fracture flow and the divergence of the one-dimensional fracture flux.A zero-dimensional pressure(point element)is used to express conservation of mass where fractures intersect.The issuing simulation is then reduced using the MHM method leading to accurate results with a very reduced number of global equations.A general system was developed where fracture geometries and conductivities are specified in an input file and meshes are generated using the public domain mesh generator GMsh.Several test cases illustrate the effectiveness of the proposed approach by comparing the multiscale results with direct simulations.

A Mixed Formulation of Stabilized Nonconforming Finite Element Method for Linear Elasticity

Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement.Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement.The mixed method is locking-free,i.e.,the convergence does not deteriorate in the nearly incompressible or incompressible case.The optimal convergence order is shown in the L^(2)-norm for stress and in the broken H1-norm and L2-norm for displacement,respectively.Finally,some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain(DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients.The semi-and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface,where the arbitrary Lagrangian-Eulerian(ALE)technique is employed to deal with the moving and immersed subdomain.Stability and optimal convergence properties are obtained for both schemes.Numerical experiments are carried out for different scenarios of jump coefficients,and all theoretical results are validated.

A Stabilized Crank-Nicolson Mixed Finite Element Method for Non-stationary Parabolized Navier-Stokes Equations

In this study, a time semi-discrete Crank-Nicolson （CN） formulation with second-order time accu- racy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element （SCNMFE） formulation based on two local Gauss integrals and parameter- free with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.

STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES

Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes.Two kinds of penalty terms are introduced in the stabilized mixed formulation,which are the jump penalty term for the displacement and the divergence penalty term for the stress.We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress,where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation.The stabilized mixed method is locking-free.The optimal convergence order is derived in the L^(2)-norm for stress and in the broken H^(1)-norm and L^(2)-norm for displacement.A numerical test is carried out to verify the optimal convergence of the stabilized method.

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

二格子的混合有限元素方法为不稳定的反应散开方程基于向后的 Euler 计划被建议。计划由为速度和压力使用最低的相等顺序的对与稳定的混合有限元素计划结合。二格子的方法也是的空间使用了到还原剂耗时。这条途径的好处是避免更高的衍生物，但是有更多的有利稳定性，并且同时得到二个未知变量的数字解决方案。稳定性分析和错误估计在这个工作被给。最后，理论结果被数字例子验证。

A REDUCED MFE FORMULATION BASED ON POD FOR THE NON-STATIONARY CONDUCTION-CONVECTION PROBLEMS

In this article, a reduced mixed finite element （MFE） formulation based on proper orthogonal decomposition （POD） for the non-stationary conduction-convection problems is presented. Also the error estimates between the reduced MFE solutions based on POD and usual MFE solutions are derived. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced MFE formulation based on POD is feasible and efficient in finding numerical solutions for the non-stationary conduction-convection problems.

A REDUCED-ORDER MFE FORMULATION BASED ON POD METHOD FOR PARABOLIC EQUATIONS

In this paper, we extend the applications of proper orthogonal decomposition （POD） method, i.e., apply POD method to a mixed finite element （MFE） formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.

Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations

The two-level penalty mixed finite element method for the stationary Navier-Stokes equations based on Taylor-Hood element is considered in this paper. Two algorithms are proposed and analyzed. Moreover, the optimal stability analysis and error estimate for these two algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms.

Finite element method based on combination of "saddle point" variational formulations

A modified mixed/hybrid finite element method, which is no longer required to satisfy the Babuska-Brezzi condition, is referred to as a stabilized method Based on the duality of vanational principles in solid mechanics, a new type of stabilized method, called the combinatorially stabilized mixed/hybrid finite element method, is presented by weight-averaging both the primal and the dual "saddle-point" schemes. Through a general analysis of stability and convergence under an abstract framework, it is shown that for the methods only an inf-sup inequality much weaker than Babuska-Brezzi condition needs to be satisfied. As a concrete application, it is concluded that the combinatorially stabilized Raviart and Thomas mixed methods permit the C -elements to replace the H(div; Ω)-elements.

OPTIMAL MIXED H-P FINITE ELEMENT METHODS FOR STOKES AND NON-NEWTONIAN FLOW

Presents an h -- p finite element methods based upon a mixed variational formulation for the three-field Stokes equations and linearized Non-Newtonian flow. Computation of the algebraic system generated from Problem H[sub h]; Methodology; Results and discussion.

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

In this paper,we develop a two-grid method(TGM)based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations.A two-grid algorithm is proposed for solving the nonlinear system,which consists of two steps:a nonlinear FE system is solved on a coarse grid,then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution.The fully discrete numerical approximation is analyzed,where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))in L^(2)-norm are presented for two-grid scheme,where t,H and h are the time step size,coarse grid mesh size and fine grid mesh size,respectively.Finally,numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

基于三维有限元的DCM平面布置对边坡稳定的影响

在应用于边坡稳定问题时,DCM(深层水泥搅拌桩)的平面布置形式直接关系到其受力的合理性,进而影响边坡的稳定和安全。针对当前工程实践中采用不同DCM平面布置形式解决边坡稳定问题时,其受力机理仍存在一定程度的认识不足,基于三维有限元方法,结合不同的DCM平面布置形式(桩式、横墙壁式、纵墙壁式、格栅式),对加固后边坡从整体稳定、位移和应力等方面进行分析,探讨采用不同DCM平面布置形式加固边坡后的位移和应力分布规律及特点。研究表明:纵墙壁式和格栅式布置相对于桩式布置,边坡整体稳定安全系数显著提高,位移和应力明显减小且分布更为合理,其中纵墙壁式布置优势最为明显。