Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

A conservative compact difference scheme for the coupled Klein–Gordon–Schrödinger equation

In this article, a conservative compact difference scheme is presented for the periodic initial‐value problem of Klein–Gordon–Schrödinger equation. On the basis of some inequalities about norms and the priori estimates, convergence of the difference solution is proved with order O(h⁴ +τ ²) in maximum norm. Numerical experiments demonstrate the accuracy and efficiency of the compact scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Superconvergence analysis of a linearized three‐step backward differential formula finite element method for nonlinear Sobolev equation

A linearized three‐step backward differential formula (BDF) Galerkin finite element method (FEM) is developed for nonlinear Sobolev equation with bilinear element. Temporal error and spatial error are discussed through introducing a time‐discrete system. Solutions of the time‐discrete system are bounded in H²‐norm by the temporal error. Superconvergence results of order O(h² + τ³) in H¹‐norm for the original variable are deduced based on the spatial error. Some new tricks are utilized to get higher order of the temporal error and the spatial error. At last, two numerical examples are provided to support the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.     

Solving the equation −uxx − ϵuyy = f(x,y, u) by an O(h4) finite difference method

The semi‐linear equation −u_xx − ϵu_yy = f(x, y, u) with Dirichlet boundary conditions is solved by an O(h⁴) finite difference method, which has local truncation error O(h²) at the mesh points neighboring the boundary and O(h⁴) at most interior mesh points. It is proved that the finite difference method is O(h⁴) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000     

Compact difference schemes for heat equation with Neumann boundary conditions (II)

This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) 25 (2009), 1320–1341]. A different compact difference scheme for the one‐dimensional linear heat equation is developed. Truncation errors of the proposed scheme are O(τ² + h⁴) for interior mesh point approximation and O(τ² + h³) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600–616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949–959) and Dai (NMPDE 27 (2011), 436–446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ² + h⁴) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009), 1320–1341), where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006), 600–616) is only O(τ² + h^3.5) while the numerical accuracy is O(τ² + h⁴), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949–959) is O(τ² + h^2.5), while the actual numerical accuracy is O(τ² + h³). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O(τ² + h⁴) and O(τ² + h³), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O(τ² + h⁴) is proved. A compact ADI difference scheme for solving two‐dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation(NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 ε≤ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(ε~2) and O(1) in time and space,respectively. We begin with the conservative Crank-Nicolson finite difference(CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates.     

Planar binary trees and perturbative calculus of observables in classical field theory

We study the Klein–Gordon equation coupled with an interaction term (□+m²)φ+λφ^p=0. In the linear case (λ=0) a kind of generalized Noether's theorem gives us a conserved quantity. The purpose of this paper is to find an analogue of this conserved quantity in the interacting case. We will see that we can do this perturbatively, and we define explicitly a conserved quantity, using a perturbative expansion based on Planar Trees and a kind of Feynman rule. Only the case p=2 is treated but our approach can be generalized to any ^p-theory.     

Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrödinger equation perturbed by wave operator

This study is devoted to analysis of semi-implicit compact finite difference (SICFD) methods for the nonlinear Schrödinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter ε∈(0,1]$\epsilon \in (0,1]$. Uniform l^∞$l^{\infty }$-norm error bounds of the proposed SICFD schemes are built to give immediate insight on point-wise error occurring as time increases, and the explicit dependence of the mesh size and time step on the parameter ε is also figured out. In the small ε   regime, highly oscillations arise in time with O(ε²)$O({\epsilon }^2)$-wavelength. This highly oscillatory nature in time as well as the difficulty raised by the compact FD discretization make establishing the l^∞$l^{\infty }$-norm error bounds uniformly in ε   of the SICFD methods for NLSW to be a very interesting and challenging issue. The uniform l^∞$l^{\infty }$-norm error bounds in ε   are proved to be of O(h⁴+τ)$O(h^4+\tau )$ and O(h⁴+τ^2/3)$O(h^4+{\tau }^{2/3})$ with time step τ and mesh size h for well-prepared and ill-prepared initial data. Finally, numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes.     

A fully discrete spectral method for fractional Cattaneo equation based on Caputo–Fabrizo derivative

Recently Caputo and Fabrizio introduced a new derivative with fractional order without singular kernel. The derivative can be used to describe the material heterogeneities and the fluctuations of different scales. In this article, we derived a new discretization of Caputo–Fabrizio derivative of order α (1 < α < 2) and applied it into the Cattaneo equation. A fully discrete scheme based on finite difference method in time and Legendre spectral approximation in space is proposed. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in H¹ norm is O(τ² + N^1?m), where τ, N and m are the time‐step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Furthermore, the accuracy and applicability of the scheme are confirmed by numerical examples to support the theoretical results.     

Superconvergence analysis and extrapolation of quasi-Wilson nonconforming finite element method for nonlinear Sobolev equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa-tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that (▽h ( u-Ihu )1, ▽hvh) h may be estimated as order O ( h2 ) when u ∈ H3 (Ω), where Ihu denotes the bilinear interpolation of u , vh is a polynomial belongs to quasi-Wilson finite element space and ▽h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O ( h2 ) /O ( h3 ) in broken H 1-norm, which is one/two order higher than its interpolation error when u ∈ H3 (Ω) /H4 (Ω). Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O ( h3 ), two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.     

Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> (<Emphasis Type="Italic">r</Emphasis> &gt; 1)

Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ?(τ): = h(τ)|sin(τ/2)|^?1 g(|sin(τ/2)|) (τ ∈ ?), where g(t) is a concave modulus of continuity slowly changing at zero such that t ^?1 g(t) ∈ L ¹[0, 1] and h(τ) is a positive function from the class C _2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [?1, 1].     

A class of difference scheme for solving telegraph equation by new non-polynomial spline methods

In this paper, by using a new non-polynomial parameters cubic spline in space direction and compact finite difference in time direction, we get a class of new high accuracy scheme of O(τ⁴ + h²) and O(τ⁴ + h⁴) for solving telegraph equation if we suitably choose the cubic spline parameters. Meanwhile, stability condition of the difference scheme has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme.     

A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations

In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is O(h⁴+τ⁵). It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method.     

Error bounds for multistep methods revisited

For linear multistep methods with constant stepsize we consider error bounds in terms of weightedL ₂-norms ofh ^px^(p) rather than ofh ^px^(p+1). The bounds apply to stiff systemsx'=Ax+f(t,x) where the spectrum ofA lies in a sector andf is of moderate size.     

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein–Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter 0 < e << 1{0 < {\varepsilon}\ll 1} which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i.e. there are propagating waves with wavelength of O(e²){O({\varepsilon}^2)} and O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter e{{\varepsilon}}. Based on the error bounds, in order to compute ‘correct’ solutions when 0 < e << 1{0 < {\varepsilon}\ll1}, the four FDTD methods share the same e{{\varepsilon}}-scalability: t = O(e³){\tau=O({\varepsilon}^3)}. Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their e{{\varepsilon}} -scalability is improved to τ = O(1) and t = O(e²){\tau=O({\varepsilon}^2)} for linear and nonlinear KG equations, respectively, when 0 < e << 1{0 < {\varepsilon}\ll1}. Numerical results are reported to support our error estimates.     

Higher order representations of the Robbins–Monro process

For quasi-linear regression functions, the Robbins–Monro process X_n is decomposed in a sum of a linear form and a quadratic form both defined in the observation errors. Under regularity conditions, the remainder term is of order O(n^−3/2) with respect to the L_p-norm. If a cubic form is added, the remainder term can be improved up to an order of O(n⁻²). As a corollary the expectation of X_n is expanded up to an error of order O(n⁻²). This is used to correct the bias of X_n up to an error of order O(n^−3/2 log n).     

Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations

The coupled nonlinear Schrödinger–Boussinesq (SBq) equations describe the nonlinear development of modulational instabilities associated with Langmuir field amplitude coupled to intense electromagnetic wave in dispersive media such as plasmas. In this paper, we present a conservative compact difference scheme for the coupled SBq equations and analyze the conservative property and the existence of the scheme. Then we prove that the scheme is convergent with convergence order O(τ² + h⁴) in L_∞‐norm and is stable in L_∞‐norm. Numerical results verify the theoretical analysis.     

Equidistribution of Heegner points and the partition function

Let p(n) denote the number of partitions of a positive integer n. In this paper we study the asymptotic growth of p(n) using the equidistribution of Galois orbits of Heegner points on the modular curve X ₀(6). We obtain a new asymptotic formula for p(n) with an effective error term which is O(n^-(\frac12+d)){O(n^{-(\frac{1}{2}+\delta)})} for some δ > 0. We then use this asymptotic formula to sharpen the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n).     

Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation

In this paper, we propose a compact finite difference scheme for computing the Klein–Gordon–Schrödinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O(h⁴+τ²)$O(h^4+{\tau }^2)$ in the l^∞$l^{\infty }$-norm with time step τ and mesh size h. Finally, numerical results are reported to test the theoretical results.     

A Priori Error Estimates for Finite Element Methods for H (2,1)-Elliptic Equations

The convergence of finite element methods for linear elliptic boundary value problems of second and forth order is well understood. In this article, we introduce finite element approximations of some linear semi-elliptic boundary value problem of mixed order on a two-dimensional rectangular domain Q. The equation is of second order in one direction and forth order in the other and appears in the optimal control of parabolic partial differential equations if one eliminates the control and the state (or the adjoint state) in the first order optimality conditions. We establish a regularity result and estimate for the finite element error of conforming approximations of this equation. The finite elements in use have a tensor product structure, in one dimension we use linear, quadratic or cubic Lagrange elements in the other dimension cubic Hermite elements. For these elements, we prove the error bound O(h ² + τ ^k ) in the energy norm and O((h ² + τ ^k )(h ² + τ)) in the L ²(Q)-norm.