Yali Gao

A stabilized finite element method based on two local Gauss integrations for a coupled Stokes-Darcy problem

In this paper, a stabilized mixed finite element method for a coupled steady Stokes-Darcy problem is proposed and investigated. This method is based on two local Gauss integrals for the Stokes equations. Its originality is to use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the coupled Stokes-Darcy problem by using the lowest equal-order finite element triples. This new method has several attractive computational features: parameter free, flexible, and altering the difficulties inherited in the original equations. Stability and error estimates of optimal order are obtained by using the lowest equal-order finite element triples ( P 1 - P 1 - P 1 ) and ( Q 1 - Q 1 - Q 1 ) for approximations of the velocity, pressure, and hydraulic head. Finally, a series of numerical experiments are given to show that this method has good stability and accuracy for the coupled problem with the Beavers-Joseph-Saffman-Jones and Beavers-Joseph interface conditions.

A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation

A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method.

A time-splitting Galerkin finite element method for the Davey–Stewartson equations

Abstract In this paper, we propose a time-splitting Galerkin method to investigate the evolution of the Davey–Stewartson equations. Using time-splitting, we split the nonlinear equations into two subequations: a linear equation and a nonlinear equation. The nonlinear equation is directly solved and the linear equation is approximated by Galerkin finite element. The error norms and conservation variables of four numerical experiments illustrate the efficiency and accuracy of our numerical scheme.

Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn--Hilliard--Navier--Stokes--Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn--Hilliard--Navier--Stokes equations in the free flow region and Cahn--Hilliard--Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability o...

Discontinuous Finite Volume Element Method for a Coupled Non-stationary Stokes–Darcy Problem

In this paper, a discontinuous finite volume element method was presented to solve the nonstationary Stokes–Darcy problem for the coupling fluid flow in conduits with porous media flow. The proposed numerical method is constructed on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by piecewise constant elements. The unique solvability of the approximate solution for the discrete problem is derived. Optimal error estimates of the semi-discretization and full discretization with backward Euler scheme in standard

Galerkin finite element methods for two-dimensional RLW and SRLW equations

L^2

A decoupled scheme with leap-frog multi-time step for non-stationary Stokes–Darcy system

L2-norm and broken