Error Estimates for the Finite Element Solutions of Some Variational Inequalities with Nonlinear Monotone Operator

In this paper, we estimate the error of the linear finite element solutions of the obstacle problem and the unilateral problem with monotone operator. We obtained $O(h)$ error bound for the obstacle problem and $O(h^{3/4})$ error bound for the unilateral problem. And if the solution $u^*$ of the unilateral problem possesses more smoothness, then $O(h)$ error bound can be obtained in the same way as [2].     

Optimal Interior and Local Error Estimates of a Recovered Gradient of Linear Elements on Nonuniform Triangulations

1.IntroductionConsideramodelellipticboundaryvalueproblemwhereflCRd,d=1,2,3,isaboundedpolyhedraldomainwithaLip8chitzboundaryfEL'(fl),aijareLipschitz-continuousfunctionsandthematrixA=(aij)issymmetricanduniformlypositivedefinitewithrespecttoxEn.Itisknownthatthefiniteelementmethodappliedto(1.1)mayproducesomesuperconvergencephenomenaeveniftheusedmeshesarenonuniform[5'8'9'1o'121.InarecentpaPer[61,aninteriorerrorestimatefortherecoveredgradientofGalerkinpiecewiselinearaPproximationshasbeenproposed…     

A Fourth Order Finite Difference Approximation to the Eigenvalues Approximation to the Eigenvalues of a Clamped Plate

In a 21-point finite difference scheme, assign suitable interpolation values to the fictitious node points. The numerical eigenvalues are then of $O(h^2)$ precision. But the corrected value $\hat{λ}_h=λ_h+\frac{h^2}{6}λ_h^{\frac{3}{2}}$ and extrapolation $\hatλ_h=\frac{4}{3}λ_{\frac{λ}{2}}-\frac{1}{3}λ_h$can be proved to have $O(h^4)$ precision.     

Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay

In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.     

Finite element algorithm based on high-order time approximation for time fractional convection-diffusion equation

In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result $O(h^{r+1}+\tau^{3-\alpha})$ can be derived, which illustrates that time convergence rate is higher than the order $(2-\alpha)$ derived by $L1$-approximation. Finally, to validate our theoretical results, some computing data are provided.     

半导体瞬态问题的修正迎风有限体积格式

半导体器件的瞬时状态由包含三个拟线性偏微分方程所组成的方程组的初边值问题来描述.其中电子位势方程足椭圆型的,电子和空穴浓度方程是对流扩散型的.对电子位势方程采用一次元有限体积法米逼近,对电子浓度和空穴浓度方程采用修正的迎风有限体积方法来逼近,并进行详细的理论分析,关于位势得到O(h Δt)阶的H1模误差估计结果,关于浓度得到O(h2 Δt)阶的L2模误差估计结果.最后,给出数值例子.     

A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.     

A Trilayer Difference Scheme for One-Dimensional Parabolic Systems

In order to obtain the numerical solution for a one-dimensional parabolic system, an unconditionally stable difference method is investigated in [1]. If the number of unknown functions is M, for each time step only M times of calculation are needed. The rate of convergence is $O(\tau+h^2)$. On the basis of [1], an alternating calculation difference scheme is presented in [2]; the rate of the convergence is $O(\tau^2+h^2)$. The difference schemes in [1] and [2] are economic ones. For the $\alpha$-$th$ equation, only $U_{\alpha}$ is an unknown function; the others $U_{\beta}$ are given evaluated either in the last step or in the present step. So the practical calculation is quite convenient. The purpose of this paper is to derive a trilayer difference scheme for one-dimensional parabolic systems. It is known that the scheme is also unconditionally stable and the rate of convergence is $O(\tau^2+h^2)$.     

A compact difference scheme for fourth-order fractional sub-diffusion equations with Neumann boundary conditions

In this paper, a compact finite difference scheme with global convergence order $O(\tau^{2}+h^4)$ is derived for fourth-order fractional sub-diffusion equations subject to Neumann boundary conditions. The difficulty caused by the fourth-order derivative and Neumann boundary conditions is carefully handled. The stability and convergence of the proposed scheme are studied by the energy method. Theoretical results are supported by numerical experiments.     

The Calahan Method for Parabolic Equations with Time-Dependent Coefficients

A modified Calahan method for parabolic equations with time-dependent coefficients is presented. It is shown that the convergence order is $O(h^{r+1}+k^3)$ while the convergence order obtained in [1] for a standard Calahan method is only $O(h^{r+1}+k^2)$.     

带非局部边界条件的非线性抛物方程的隐式差分解法

在准静态弹性力学中常遇到求解带有非局部边界条件的抛物方程初边值问题.本文构造了一个数值求解带有非局部边界条件的非线性抛物方程的隐式差分格式,利用离散泛函分析的知识和不动点定理证明了差分解是存在的,且在离散最大模意义下关于时间步长一阶收敛,关于空间步长二阶收敛,并给出了数值算例.     

A numerical scheme based on mean value solutions for the helmholtz equation on triangular grids

A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.

     

Extrapolation of Finite Element Approximation in a Rectangular Domain

Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution $u$ but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0＜\alpha＜1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.     

Extrapolation of Nyström Solutions of Boundary Integral Equations on Non-Smooth Domains

The interior Dirichlet problem for Laplace's equation on a plane polygonal region $\Omega$ with boundary $\Gamma$ may be reformulated as a second kind integral equation on $\Gamma$. This equation may be solved by the Nyström method using the composite trapezoidal rule. It is known that if the mesh has $O(n)$ points and is graded appropriately, then $O(1/n^2)$ convergence is obtained for the solution of the integral equation and the associated solution to the Dirichlet problem at any $x\in \Omega$. We present a simple extrapolation scheme which increases these rates of convergence to $O(1/n^4)$ .     

Analysis of recovery type a posteriori error estimators for mildly structured grids

Some recovery type error estimators for linear finite elements are analyzed under 0)$"> regular grids. Superconvergence of order is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.

     

Acceleration of the Convergence in Finite Difference Method by Predictor-Corrector and Splitting Extrapolation Methods

Two types of combination methods for accelerating the convergence of the finite difference method are presented. The first is based on an interpolation principle (correction method) and the second one on extrapolation principle. They improve the convergence form $O(h^2)$ to $O(h^4)$. The main advantage when compared with standard methods, is that the computational work can be split into independent parts, which can be carried out in parallel.     

A PRACTICAL PARALLEL DIFFERENCE SCHEME FOR PARABOLIC EQUATIONS

A practical parallel difference scheme for parabolic equations is constructed as follows: to decompose the domain Ω into some overlapping subdomains, take flux of the last time layer as Neumann boundary conditions for the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme on each subdomain, then take correspondent values of its neighbor subdomains as its values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. The scheme is unconditionally convergent. Though its truncation error is O(τ h), the convergent order for the solution can be improved to O(τ h2).     

A high order difference method for fractional sub-diffusion equations with the spatially variable coefficients under periodic boundary conditions

In this paper, we propose a difference scheme with global convergence order $O(\tau^{2}+h^4)$ for a class of the Caputo fractional equation. The difficulty caused by the spatially variable coefficients is successfully handled. The unique solvability, stability and convergence of the finite difference scheme are proved by use of the Fourier method. The obtained theoretical results are supported by numerical experiments.     

Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0 < \varepsilon≤ 1.$ In fact, the solution to this equation propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon≪1,$ which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at $O(\tau^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon ∈ (0,1]$ with $\tau > 0$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(\tau)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $ε=O(1)$ or $0<ε≤\tau.$ Thus the meshing strategy requirement (or $ε$-scalability) of the two MTIs is $\tau =O(1)$ for $0<ε≪1,$ which is significantly improved from $\tau =O(ε^3)$ and $\tau =O(ε^2)$ requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.     

Analysis of a bilinear finite element for shallow shells. II: Consistency error

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound can be obtained if the contribution of transverse shear is neglected. Here is the thickness of the shell, the mesh spacing, and a smoothness parameter. In the bending-dominated case, the uniformly optimal bound is achievable but requires that membrane and transverse shear strains are of order as . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound .