Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers

This paper presents an anisotropic adaptive finite element method （FEM） to solve the governing equations of steady magnetohydrodynamic （MHD） duct flow. A resid- ual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu es- timates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.

CONSERVATIVE CONFORMING AND NONCONFORMING VEMS FOR FOURTH ORDER NONLINEAR SCHRODINGER EQUATIONS WITH TRAPPED TERM

This paper aims to construct and analyze the conforming and nonconforming virtual element methods for a class of fourth order nonlinear Schrodinger equations with trapped term.We mainly consider three types of virtual elements,including H^(2) conforming virtual element,C^(0) nonconforming virtual element and Morley-type nonconforming virtual element.The fully discrete schemes are constructed by virtue of virtual element methods in space and modified Crank-Nicolson method in time.We prove the mass and energy conservation,the boundedness and the unique solvability of the fully discrete schemes.After introducing a new type of the Ritz projection,the optimal and unconditional error estimates for the fully discrete schemes are presented and proved.Finally,two numerical examples are investigated to confirm our theoretical analysis.

UNCONDITIONAL ERROR ANALYSIS OF VEMS FOR A GENERALIZED NONLINEAR SCHRODINGER EQUATION

In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.

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.

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.