Error Estimates for Finite Element Approximations of a Viscous Wave Equation

We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on the energy method is developed. Optimal error estimates in both time and space are obtained. For sufficiently smooth solutions, it is demonstrated that the maximal error in the L ²-norm over a finite time interval converges optimally as O(h ^p+1 + Δt ^s ), where p denotes the polynomial degree, s = 1 or 2, h the mesh size, and Δt the time step.     

Multistep characteristic method for incompressible flow in porous media

In this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media. The discretization involves a three-level time scheme based on the characteristic method and a five-point finite difference scheme for space discretization. We prove that the convergence is of order O(h²+(Δt)²), which is in contrast to the convergence of order O(h+Δt) proved for a singlestep characteristic with the same space discretization. Numerical experiments demonstrate the stability and second-order convergence of the scheme.     

Optimal l~∞ error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions

Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h~2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.     

A new high accuracy non-polynomial tension spline method for the solution of one dimensional wave equation in polar co-ordinates

In this paper, we propose a new three-level implicit nine point compact finite difference formulation of O(k² + h⁴) based on non-polynomial tension spline approximation in r-direction and finite difference approximation in t-direction for the numerical solution of one dimensional wave equation in polar co-ordinates. We describe the mathematical formulation procedure in details and also discuss the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.     

Estimates of derivatives of random functions I

If one looks for an optimal (by criterion of minimal variance) linear estimate of s′(t) from the observation of u(t) = s(t) + n(t), where n(t) is noise and s(t) is useful signal, then one can derive an integral equation for the weight function of optimal estimate. This integral equation is often difficult to solve and, even if one can solve it, it is difficult to construct the corresponding filter. In this paper an optimal estimate of s′ on a subset of all linear estimates in sought and it is shown that this quasioptimal estimate is easy to calculate, the corresponding filter is easy to construct, and the error of this estimate differs little from the error of optimal estimates. It is also shown that among all estimates (linear and nonlinear) of s′ for ∥n∥ ? δ and ∥s″∥ ? M the best estimate is given by Δ_hu = (2h)^?1 [u(t + h) ? u(t ? h)] with $\text{h = (}\text{2δ}\text{M}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

A New Compact Finite Difference Method for Solving the Generalized Long Wave Equation

The aim of this article is to analyze a new compact finite difference method (CFDM) for solving the generalized regularized long wave (GRLW) equation. This method leads to a system of linear equations involving tridiagonal matrices and the rate of convergence of the method is of order O(k ² + h ⁴) where k and h are mesh sizes of time and space variables, respectively. Stability analysis of the method is investigated by the energy method and an error estimate is given. The propagation of single solitons and interaction of two solitary waves are applied to validate the method which is found to be accurate and efficient. Three invariants of the motion are evaluated to determine conservation properties of the method.     

An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(τ² + h⁶) accurate, where τ, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.     

Some superconvergence results of high-degree finite element method for a second order elliptic equation with variable coefficients

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x ₀ for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h ^p+1 |ln h|²) and O(h ^p+2 |ln h|²) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination of the weak estimates and local symmetric technique is also effective for superconvergence analysis of the second order elliptic equation with variable coefficients.     

Asymptotic stability of a class of integro-differential equations

We examine the asymptotic stability of the zero solution of the first-order linear equation x′(t) = Ax(t) + ∝₀^tB(t ? s) x(s) ds, where B(t) is integrable and does not change sign on [0, ∞). The results are applied to an examination of the stability of equilibrium of some nonlinear population models.     

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h ^s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H ^1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h ^3/2?ε) with ε > 0 if u ∈ H ³(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.     

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.     

Finite element methods for symmetric hyperbolic equations

A finite difference method for the solution of symmetric positive differential equations has already been developped (Katsanis [4]). The finite difference solutions where shown to converge at the rateO(ith ^1/2) ash approaches zero,h being the maximum distance between two adjacent mesh points. Here we try to get a better rate of convergence, using a Rayleigh Ritz Galerkin method. We first give a “weak” formulation of the equations, slightly different from the usual one (Friedrichs [3]), in order to take into account the boundary conditions. We define a finite dimensional subspaceV _h ofH ¹(Ω), in which we look for an approximate solutionu _h . We show that when the exact solutionu is smooth enough, we get the error estimate: $$\left| {u - u_h } \right|L^2 (\Omega ) \leqq C\mathop {\inf }\limits_{v_h \in V_h } \left\{ {\left\| {u - v_h } \right\|H^1 (\Omega ) + \mathop {\sup }\limits_{w_h \in V_h } \frac{{\int\limits_\Gamma {\left| {u - v_h } \right|\left| {w_h } \right|d\Gamma } }}{{\left| {w_h } \right|L^2 (\Omega )}}} \right\}$$ where |·| denotes the Euclidean norm inR ^P . Thus, as is the case for elliptic or parabolic equations, the problem of estimating the error is reduced to questions in approximation theory. When those results are applied to finite element methods, with polynomial approximations of degree ≦k over eachn-simplex we obtain a rate of convergence ofO(h ^k) ash approaches zero,h being the supremum of the diameters of then-simplices.     

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H¹-norm is proved to be O(h+H³|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.     

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H ¹-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\) (h ² t ^?1/2) for positive time are established assuming the initial function p ₀ ∈ H ²(Ω) ∩ H ₀ ¹ (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\) (h ² t ^?1) when p ₀ is only in H ₀ ¹ (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.     

Uniformly Controllable Schemes for the Wave Equation on the Unit Square

The paper deals with the numerical approximation of the HUM control of the 2D wave equation. Most of the discrete models obtained with classical finite difference or finite element methods do not produce convergent sequences of discrete controls, as the mesh size h and the time step Δt go to zero. We introduce a family of fully-discrete schemes, nondispersive, stable under the condition \(\Delta t\leq h\slash\sqrt{2}\) and uniformly controllable with respect to h and Δt. These implicit schemes differ from the usual explicit one (obtained with leapfrog time approximation and five point spatial approximations) by the addition of terms proportional to h ² and Δt ². Numerical experiments for nonsmooth initial conditions on the unit square using a conjugate gradient algorithm indicate the excellent performance of the schemes.     

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order 0 < α < 1 ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent withO(Τ +h ²), where Τ andh are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.     

Compact finite difference scheme for the solution of time fractional advection-dispersion equation

In this paper, a compact finite difference method is proposed for the solution of time fractional advection-dispersion equation which appears extensively in fluid dynamics. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of order O(τ ^2???α ), 0?<?α?<?1, and spatial derivatives are replaced with a fourth order compact finite difference scheme. We will prove the unconditional stability and solvability of proposed scheme. Also we show that the method is convergence with convergence order O(τ ^2???α ?+?h ⁴). Numerical examples confirm the theoretical results and high accuracy of proposed scheme.     

Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)^3?α ), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^3?α ), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis.