A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson Scheme

A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson scheme is considered in this paper. The method combines the Crank–Nicolson scheme with a stabilized finite element method which uses the lowest equal-order element pair, i.e., the stabilized finite element method is applied for the spatial approximation and the time discretization is based on the Crank–Nicolson scheme. Moreover, we present optimal error estimates and prove that the scheme is unconditionally stable and convergent. Finally, numerical tests confirm the theoretical results of the presented method.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces

We develop an unconditionally stable direct discretization scheme for solving the phase-field crystal equation on surfaces. The surface is discretized by using an unstructured triangular mesh. Gradient, divergence, and Laplacian operators are defined on triangular meshes. The proposed numerical method is second-order accurate in space and time. At each time step, the proposed computational scheme results in linear elliptic equations to be solved, thus it is easy to implement the algorithm. It is proved that the proposed scheme satisfies a discrete energy-dissipation law. Therefore, it is unconditionally stable. A fast and efficient biconjugate gradients stabilized solver is used to solve the resulting discrete system. Numerical experiments are conducted to demonstrate the performance of the proposed algorithm.     

Bubble stabilized discontinuous Galerkin method for parabolic and elliptic problems

In this paper we give an analysis of a bubble stabilized discontinuous Galerkin method for elliptic and parabolic problems. The method consists of stabilizing the numerical scheme by enriching the discontinuous affine finite element space elementwise by quadratic bubbles. This approach leads to optimal convergence in the space and time discretization parameters.     

An optimization-based numerical method for diffusion problems with sign-changing coefficients

A new optimization-based numerical method is proposed for the solution to diffusion problems with sign-changing conductivity coefficients. In contrast to existing approaches, our method does not rely on the discretization of a stabilized equation, and the convergence of the scheme can be proved without any symmetry assumption on the mesh near the interface where the conductivity sign changes.     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

Dynamical analysis of Kirchhoff plates by an explicit time integration Morley element method

An explicit time integration finite element method is proposed to investigate dynamical analysis of Kirchhoff plates, where the Morley element is used for spatial discretization and the second-order central scheme for time discretization. Certain error estimates in the energy norm are achieved. A number of numerical results are included to show computational performance of the method.     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Fully Discrete Error Estimation by the Method of Lines for a Nonlinear Parabolic Problem

A posteriori error estimates for a nonlinear parabolic problem are introduced. A fully discrete scheme is studied. The space discretization is based on a concept of hierarchical finite element basis functions. The time discretization is done using singly implicit Runge-Kutta method (SIRK). The convergence of the effectivity index is proven.     

两个带变时间步的两重网格算法的稳定性分析

对用于求解非线性发展方程的两个带变时间步的两重网格算法，对空间变量用有限元离散，对时间变量分别用一阶精度Euler显式和二阶精度半隐式差分格式离散，然后构造两重网格算法，通过深入的稳定性分析，得出本文的算法优于标准全离散有限元算法。     

Error estimates for a stabilized finite element method for the Oldroyd B model

In this paper, we study a new approximation scheme of transient viscoelastic fluid flow obeying an Oldroyd-B-type constitutive equation. The new stabilized formulation bases on the choice of a modified Euler method connected to the streamline upwinding Petrov-Galerkin (SUPG) method [M. Bensaada, D. Esselaoui, D. Sandri, Stabilization method for continuous approximation of transient convection problem, Numer. Methods Partial Differential Equations 21 (2004) 170-189], in order to stabilize the tensorial transport term of the Oldroyd derivative. Suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution. A priori error estimates for the approximation in terms of the mesh parameter h and the time discretization parameter Δt are derived.     

A higher order uniformly convergent method for singularly perturbed parabolic turning point problems

In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection-diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second-order convergence up to a logarithmic factor in space and second-order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes.     

A monotone scheme for Hamilton–Jacobi equations via the nonstandard finite difference method

A usual way of approximating Hamilton–Jacobi equations is to couple space finite element discretization with time finite difference discretization. This classical approach leads to a severe restriction on the time step size for the scheme to be monotone. In this paper, we couple the finite element method with the nonstandard finite difference method, which is based on Mickens' rule of nonlocal approximation. The scheme obtained in this way is unconditionally monotone. The convergence of the new method is discussed and numerical results that support the theory are provided. Copyright © 2009 John Wiley & Sons, Ltd.     

A ROBUST DISCRETIZATION OF THE REISSNER-MINDLIN PLATE WITH ARBITRARY POLYNOMIAL DEGREE

A numerical scheme for the Reissner-Mindlin plate model is proposed.The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk[SIAM J.Numer.Anal.,26(6):1276-1290,1989].The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement.The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element.The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter t.     

On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.     

Variational integrators of higher order for flexible multibody systems

In this paper we present a time stepping scheme which is based on a variational integrator. This higher-order time stepping scheme includes constraints and a viscoelastic material formulation. A variational integrator is structure-preserving which results from using a discrete variational principle. Therefore, a variational integrator always takes the form of discrete EULER-LAGRANGE equations or the equivalent position-momentum equations. In this framework, we consider the motion of a flexible rope with non-holonomic constraints by the LAGRANGE-multiplier technique. The time stepping scheme is derived from a space-time discretization of HAMILTON's principle. The space discretization is based on one-dimensional linear LAGRANGE polynomials, whereas the time discretization is based on higher-order polynomials and higher-order quadrature rules. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal m-stage Runge-Kutta schemes for steady-state solutions of hyperbolic equations

A numerical scheme is developed to find optimal parameters and time step of m-stage Runge-Kutta (RK) schemes for accelerating the convergence to -steady-state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m-stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.     

An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation

In Becker and Jentzen (2019) and Becker et al. (2017), an explicit temporal semi-discretization scheme and a space–time full-discretization scheme were, respectively, introduced and analyzed for the additive noise-driven stochastic Allen–Cahn type equations, with strong convergence rates recovered. The present work aims to propose a different explicit full-discrete scheme to numerically solve the stochastic Allen–Cahn equation with cubic nonlinearity, perturbed by additive space–time white noise. The approximation is easily implementable, performing the spatial discretization by a spectral Galerkin method and the temporal discretization by a kind of nonlinearity-tamed accelerated exponential integrator scheme. Error bounds in a strong sense are analyzed for both the spatial semi-discretization and the spatio-temporal full discretization, with convergence rates in both space and time explicitly identified. It turns out that the obtained convergence rate of the new scheme is, in the temporal direction, twice as high as existing ones in the literature. Numerical results are finally reported to confirm the previous theoretical findings.     

非定常Stokes方程的全离散稳定化有限元格式

本文研究二维非定常Stokes方程全离散稳定化有限元方法.首先给出关于时间向后一步Euler半离散格式,然后直接从该时间半离散格式出发,构造基于两局部高斯积分的稳定化全离散有限元格式,其中空间用P_1—P_1元逼近,证明有限元解的误差估计.本文的研究方法使得理论证明变得更加简便,也是处理非定常Stokes方程的一种新的途径.     

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.