CONVERGENCE ANALYSIS OF NONCONFORMING QUADRILATERAL FINITE ELEMENT METHODS FOR NONLINEAR COUPLED SCHRODINGER-HELMHOLTZ EQUATIONS

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods(FEMs)for the nonlinear coupled Schrodinger-Helmholtz equations.Optimal L^(2) and H^(1) estimates of orders O(h^(2)+τ^(2))and O(h^(2)+τ^(2))are derived respectively without any grid-ratio condition through the following two keys.One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error,which leads to optimal temporal error estimates of order O(τ^(2))in L^(2) and the broken H^(1)-norms,as well as the uniform boundness of numerical solutions in L^(∞) norm.The other is that a novel projection is utilized,which can iron out the difficulty of the existence of the consistency errors.This leads to derive optimal spatial error estimates of orders O(h^(2))in L^(2)-norm and O(h)in the broken H^(1)-norm under the H^(2) regularity of the solutions for the time-discrete system.At last,two numerical examples are provided to confirm the theoretical analysis.Here,h is the subdivision parameter,and τ is the time step.

Nonconforming Mixed FEM Analysis for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation with Time-Space Coupled Derivative

The main contents of this paper are to establish a finite element fully-discrete approximate scheme for multi-term time-fractional mixed sub-diffusion and diffusionwave equation with spatial variable coefficient,which contains a time-space coupled derivative.The nonconforming EQ^(rot)_(1)element and Raviart-Thomas element are employed for spatial discretization,and L1 time-stepping method combined with the Crank-Nicolson scheme are applied for temporal discretization.Firstly,based on some significant lemmas,the unconditional stability analysis of the fully-discrete scheme is acquired.With the assistance of the interpolation operator I_(h)and projection operator Rh,superclose and convergence results of the variable u in H^(1)-norm and the flux~p=k_(5)(x)ru(x,t)in L^(2)-norm are obtained,respectively.Furthermore,the global superconvergence results are derived by applying the interpolation postprocessing technique.Finally,the availability and accuracy of the theoretical analysis are corroborated by experimental results of numerical examples on anisotropic meshes.

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.

A LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHOD FOR NON-STATIONARY INCOMPRESSIBLE MAGNETOHYDRODYNAMICS SYSTEM

A low order nonconforming mixed finite element method(FEM)is established for the fully coupled non-stationary incompressible magnetohydrodynamics(MHD)problem in a bounded domain in 3D.The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure,the velocity field and the magnetic field,in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by H^(1)(Ω)-conforming finite elements,respectively.The existence and uniqueness of the approximate solutions are shown.Optimal order error estimates of L^(2)(H^(1))-norm for the velocity field,L^(2)(L^(2))-norm for the pressure and the broken L^(2)(H^(1))-norm for the magnetic field are derived.

ANISOTROPIC EQ^(ROT)_(1) FINITE ELEMENT APPROXIMATION FOR A MULTI-TERM TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION

By employing EQ^(ROT)_(1) nonconforming finite element,the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes.Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation,the mixed case contains a special time-space coupled derivative,which leads to many difficulties in numerical analysis.Firstly,a fully discrete scheme is established by using nonconforming finite element method(FEM)in spatial direction and L1 approximation coupled with Crank-Nicolson(L1-CN)scheme in temporal direction.Furthermore,the fully discrete scheme is proved to be unconditional stable.Besides,convergence and superclose results are derived by using the properties of EQ^(ROT)_(1) nonconforming finite element.What's more,the global superconvergence is obtained via the interpolation postprocessing technique.Finally,several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.