Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations

A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results show efficiency of this approach. The proposed method is also applicable to other problems in spherical geometry.

Fully Discrete Nonlinear Galerkin Methods for Kuramoto-Sivashinsky Equation and Their Error Estimates

In this paper,the uniform error estimates with respect to t∈[0, ∞ ) of the nonlinear Galerkin method are given for the long time integration of the Kuramoto-Sivashinsky equation. The nonlinear Galerkin method is used to study the asymptotic behaviour of Kuramoto-Sivashinsky equation and to construct the bifurcation diagrams.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model.The spatial discretization uses piecewise polynomials of degree k(k≥1)for the stress approximation,degree k+1 for the velocity approximation,and degree k for the numerical trace of velocity on the inter-element boundaries.The temporal discretization in the fully discrete method adopts a backward Euler difference scheme.We show the existence and uniqueness of the semi-discrete and fully discrete solutions,and derive optimal a priori error estimates.Numerical examples are provided to support the theoretical analysis.

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.

Recovery Type A Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.

Semi-Discrete and Fully Discrete Mixed Finite Element Methods for Maxwell Viscoelastic Model of Wave Propagation

Semi-discrete and fully discrete mixedfinite element methods are consid-ered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid.This mixedfinite element framework allows the use of a large class of exist-ing mixed conformingfinite elements for elasticity in the spatial discretization.In the fully discrete scheme,a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables.Error estimates of the semi-discrete and fully discrete schemes,as well as an unconditional stability result for the fully discrete scheme,are derived.Numerical experiments are provided to verify the theoretical results.

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.

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

The mixed covolume method for the regularized long wave equation is devel- oped 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.

Transition state and formation process of Stone–Wales defects in graphene

Stone–Wales(SW) defects are possibly formed in graphene and other two-dimensional materials, and have multiple influence on their physical and chemical properties. In this study, the transition state of SW defects in graphene is determined with the fully discrete Peierls theory. Furthermore, the atomic formation process is investigated by means of ab-initio simulations. The atomic structure change and energetics of the SW transformation are revealed. It is found that the transition state is at the SW bond rotation of 34.5°and the activation energy barrier is about 12 eV. This work provides a new method to investigate SW transformations in graphene-like materials and to explore unknown SW-type defects in other 2D materials.

Asymptotic-Preserving Discrete Schemes for Non-Equilibrium Radiation Diffusion Problem in Spherical and Cylindrical Symmetrical Geometries

We study the asymptotic-preserving fully discrete schemes for nonequilibrium radiation diffusion problem in spherical and cylindrical symmetric geometry.The research is based on two-temperature models with Larsen’s flux-limited diffusion operators.Finite volume spatially discrete schemes are developed to circumvent the singularity at the origin and the polar axis and assure local conservation.Asymmetric second order accurate spatial approximation is utilized instead of the traditional first order one for boundary flux-limiters to consummate the schemes with higher order global consistency errors.The harmonic average approach in spherical geometry is analyzed,and its second order accuracy is demonstrated.By formal analysis,we prove these schemes and their corresponding fully discrete schemes with implicitly balanced and linearly implicit time evolutions have first order asymptoticpreserving properties.By designing associated manufactured solutions and reference solutions,we verify the desired performance of the fully discrete schemes with numerical tests,which illustrates quantitatively they are first order asymptotic-preserving and basically second order accurate,hence competent for simulations of both equilibrium and non-equilibrium radiation diffusion problems.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite nite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane.This new class of nite elements,which is called composite nite elements,was rst introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial di erential equations on domains with complicated geometry.The aim of this paper is to introduce an effcient numerical method which gives a lower dimensional approach for solving partial di erential equations by domain discretization method.The composite nite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the ne-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the ne-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the nite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite nite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L^(∞)(L^(2))-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.