Luo Zhen-dong

DIFFERENCE SCHEME AND NUMERICAL SIMULATION BASED ON MIXED FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM

The non-stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non-stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.

A reduced second-order time accurate finite element formulation based on POD for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual second-order time accurate Crank-Nicolson finite element (CNFE) formulation for parabolic equations such that it is reduced into a second- order time accurate CNFE formulation with fewer degrees of freedom and high enough accuracy. The errors between the reduced second-order time accurate CNFE solutions and the usual second-order time accurate CNFE solutions are analyzed. It is shown by numerical examples that the reduced second-order time accurate CNFE formulation can greatly save degrees of freedom in a way that guarantees a sufficiently small errors between the reduced second-order time accurate CNFE solutions and the usual second-order time accurate CNFE solutions. The time step of the reduced second-order time accurate CNFE formulation is ten times that of the first-order time accurate reduced finite element formulation such that it could obtain very quickly the numerical solution at the moment wanted, alleviate the computer truncation error, and improve rate and accuracy in the computational process. Moreover, it is also shown that the reduced second-order time accurate CNFE formulation is feasible and efficient solving parabolic equations.

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.

A nonlinear galerkin mixed element method and a posteriori error estimator for the stationary navier-stokes equations

A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.

Mixed Finite Element Methods for the Shallow Water Equations Including Current and Silt Sedimentation (II)-The Discrete-Time Case Along Characteristics

The mixed finite element (MFE) methods for a shallow water equation system consisting of water dynamics equations, silt transport equation, and the equation of bottom topography change were derived. A fully discrete MFE scheme for the discrete-time along characteristics is presented and error estimates are established. The existence and convergence of MFE solution of the discrete current velocity, elevation of the bottom topography, thickness of fluid column, and mass rate of sediment is demonstrated.

A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations

A nonlinear Galerkin/Petrov-least squares mixed element (NGPLSME) method for the stationary Navier-Stokes equations is presented and analyzed. The scheme is that Petrov-least squares forms of residuals are added to the nonlinear Galerkin mixed element method so that it is stable for any combination of discrete velocity and pressure spaces without requiring the Babuška-Brezzi stability condition. The existence, uniqueness and convergence (at optimal rate) of the NGPLSME solution is proved in the case of sufficient viscosity (or small data).

CONVERGENCE OF SIMPLIFIED AND STABILIZED MIXED ELEMENT FORMATS BASED ON BUBBLE FUNCTION FOR THE STOKES PROBLEM

Two simplified and stabilized mixed element formats for the Stokes problem are derived by bubble function, and their convergence, i. e., error analysis, are proved. These formats can save more freedom degrees than other usual formats.

A type of new posteriori error estimators for stokes problems

In this paper, a new discrete formulation and a type of new posteriori error estimators for the second-order element discretization for Stokes problems are presented, where pressure is approximated with piecewise first-degree polynomials and velocity vector field with piecewise seconddegree polynomials with a cubic bubble function to be added. The estimators are the globally upper and locally lower bounds for the error of the finite element discretization. It is shown that the bubble part for this second-order element approximation is substituted for the other parts of the approximate solution.

Mixed finite element method of hexahedral elements for Navier-Stokes problem

In this paper, we derive a new mixed element format of hexahedral elements for Navier-Stokes problem in three-dimensional space.