A semi‐discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations

In this report, we give a semi‐discrete defect correction finite element method for the unsteady incompressible magnetohydrodynamics equations. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear magnetohydrodynamics equations is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect‐correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. In order to show the effect of our method, some numerical results are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

Error analysis of a fully discrete finite element variational multiscale method for time‐dependent incompressible Navier–Stokes equations

A finite element variational multiscale method based on two local Gauss integrations is applied to solve numerically the time‐dependent incompressible Navier–Stokes equations. A significant feature of the method is that the definition of the stabilization term is derived via two local Guass integrations at element level, making it more efficient than the usual projection‐based variational multiscale methods. It is computationally cheap and gives an accurate approximation to the quantities sought. Based on backward Euler and Crank–Nicolson schemes for temporal discretization, we derive error bounds of the fully discrete solution which are first and second order in time, respectively. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error correction iterative method for the stationary incompressible MHD flow

A new iterative finite element method for solving the stationary incompressible magnetohydrodynamics (MHD) equations is derived in this paper. The method consists of two steps at each iteration step, we need first to solve the MHD equations by the Oseen-type iterative scheme, and then an error correction strategy is applied to control the error arising from the linearization of the nonlinear MHD equations. The new method not only maintains the advantage of the standard Oseen-type scheme but also possesses a rapid rate of convergence. It is proved that the convergence rate of the proposed method is increased greatly under the uniqueness condition. The uniform stability and convergence of the new scheme are analyzed. Ample numerical experiments are performed to validate the accuracy and the efficiency of the new numerical scheme.     

A three-step Oseen-linearized finite element method for incompressible flows with damping

This paper introduces a three-step Oseen-linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen-linearized problems over a fine grid, which possess the same stiffness matrices with only various right-hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two-level method.     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

Simple finite element method in vorticity formulation for incompressible flows

A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work. Optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements. The method is convectively stable and is particularly suited for moderate to high Reynolds number flows.

     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Mixed finite element methods for incompressible flow: Stationary Stokes equations

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress‐velocity formulation. The pseudostress is approximated by the Raviart‐Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi‐optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations

In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier-Stokes equations. It focuses on the lowest equal-order velocity-pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure.     

On a finite element method for unsteady compressible viscous flows

In this article we present an analysis of a finite element method for solving two‐dimensional unsteady compressible Navier‐Stokes equations. Under the time‐stepping size restriction Δt ≤ Ch, we prove the existence and uniqueness of the numerical solution and obtain an a prior error estimate uniform in time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 152–166, 2003     

Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizon

In this article, we deal with a rigorous error analysis for the finite element solutions of the two‐dimensional Cahn–Hilliard equation with infinite time. The error estimates with respect to are proven for the fully discrete conforming piecewise linear element solution under Assumption (A1) on the initial value and Assumption (A2) on the discrete spectrum estimate in the finite element space. The analysis is based on sharp a‐priori estimates for the solutions, particularly reflecting their behavior as . Numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the fully discrete mixed finite element methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 742–762, 2017     

Finite element error analysis of a variational multiscale method for the Navier-Stokes equations

The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier–Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh. This work is partially supported by NSF grants DMS9972622, DMS20207627 and INT9814115.     

Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

Asymptotic error expansions in the sense of L ^∞-norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing technique, and the key point in deriving them is the establishment of the error estimates for the mixed regularized Green’s functions with memory terms presented in R. Ewing at al., Int. J. Numer. Anal. Model 2 (2005), 301–328. As a result of all these higher order numerical approximations, they can be used to generate a posteriori error estimators for this mixed finite element approximation. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), the Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), and Tianjin University of Finance and Economics.     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error analysis of a fully discrete projection method for magnetohydrodynamic system

In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H ( curl ) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L₂ and H 1 . The convergence rate in the L_∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

The application of iterated defect correction to theLod method for parabolic equations

The paper is concerned with the numerical solution of the initial boundary value problem for a class of multi-dimensional parabolic partial differential equations. In particular the time-integration of semi-discrete equations is investigated. An attempt is made to develop integration formulas being computationally attractive and of high accuracy, while possessing unconditional stability properties. To that end iterated defect correction is applied to the LOD method. The convergence properties of this process are investigated. Numerical experiments are reported.     

A posteriori error estimation for a defect correction method applied to conduction convection problems

We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity ν. A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a problem with known solution and the second example is a physical model of square cavity stationary flow. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

非定常BOLTZMANN方程的谱有限元方法

In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.     

Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters and are sufficiently small.