Error estimates for the Ginzburg-Landau approximation

Modulation equations play an essential role in the understanding of complicated dynamical systems near the threshold of instability. Here we look at systems defined over domains with one unbounded direction and show that the Ginzburg-Landau equation dominates the dynamics of the full problem, locally, at least over a long time-scale. As an application of our approximation theorem we look here at Bénard's problem. The method we use involves a careful handling of critical modes in the Fourier-transformed problem and an estimate of Gronwall's type.     

Error estimates for the approximation of semicoercive variational inequalities

Summary. An abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution. This condition turns out to be necessary as is demonstrated analytically and numerically. The results are applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions. For second order variational inequalities the condition is always satisfied, whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set. Applying the error estimate yields optimal order of convergence in terms of the mesh size . The numerical convergence rates observed are in good agreement with the predicted ones. Received August 16, 1993 / Revised version received March 21, 1994     

Error estimates for the approximation of multibang control problems

In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.     

Error estimates for sampling approximation of non-bandlimited functions

A new proof of the sampling theorem for non-bandlimited signals is presented. It leads to better rates of convergence for the envolved approximation process. Furthermore, the order of convergence is given for an extended class of functions.     

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193–219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L ²(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45–61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided. The authors were supported by Ministerio de Ciencia y Tecnología (Spain).     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

Error autocorrection in rational approximation and interval estimates. [A survey of results.]

The error autocorrection effect means that in a calculation all the intermediate errors compensate each other, so the final result is much more accurate than the intermediate results. In this case standard interval estimates (in the framework of interval analysis including the so-called a posteriori interval analysis of Yu. Matijasevich) are too pessimistic. We shall discuss a very strong form of the effect which appears in rational approximations to functions. The error autocorrection effect occurs in all efficient methods of rational approximation (e.g., best approxmations, Padé approximations, multipoint Padé approximations, linear and nonlinear Padé-Chebyshev approximations, etc.), where very significant errors in the approximant coefficients do not affect the accuracy of this approximant. The reason is that the errors in the coefficients of the rational approximant are not distributed in an arbitrary way, but form a collection of coefficients for a new rational approximant to the same approximated function. The understanding of this mechanism allows to decrease the approximation error by varying the approximation procedure depending on the form of the approximant. Results of computer experiments are presented. The effect of error autocorrection indicates that variations of an approximated function under some deformations of rather a general type may have little effect on the corresponding rational approximant viewed as a function (whereas the coefficients of the approximant can have very significant changes). Accordingly, while deforming a function for which good rational approximation is possible, the corresponding approximant’s error can rapidly increase, so the property of having good rational approximation is not stable under small deformations of the approximated functions. This property is “individual”, in the sense that it holds for specific functions.     

Lorentz space estimates for the Ginzburg-Landau energy

In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the “vortex-balls construction” estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space L2,∞ we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results.     

Error estimates for a galerkin approximation of a parabolic control problem

Summary Numerical approximation of a parabolic control problem with a Neumann boundary condition control is considered. The observation is the final state. The numerical approximation is based on backward discretization with respect to time and a Galerkin method in the space variables. Optimal (except for a logarithmic term) L² error estimates are derived for the optimal state. Certain error estimates for the optimal control are also given. Entrata in Redazione il 4 maggio 1977. This work was supported by the Norwegian Research Council for Science and the Humanities.     

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R ² is considered. The optimal asymptotic L₂(G) and H₁(G)-error estimates 0(h²) and 0(h) resp, are derived for a piecewise linear finite element solution.     

Error estimates for an operator-splitting method for Navier-Stokes equations: Second-order schemes

We present in this paper an error analysis of a fractional-step method for the approximation of the unsteady incompressible Navier-Stokes equations. Under mild regularity assumptions on the continuous solution, we obtain second-order error estimates in the time step size, both for velocity and pressure. Numerical results in agreement with the error analysis are also presented.     

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants are given for approximations of functions, definite integrals and indefinite integrals by means of the Sinc approximation. Although in the literature various error estimates have already been given for these approximations, those estimates were basically for examining the rates of convergence, and several constants were left unevaluated. Giving more explicit estimates, i.e., evaluating these constants, is of great practical importance, since by this means we can reinforce the useful formulas with the concept of “verified numerical computations.” In this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems: the function to be approximated is analytic in a suitable region. We also improve some formulas themselves to decrease their computational costs. Numerical examples that confirm the theory are also given.     

A new estimate for the Ginzburg-Landau approximation on the real axis

Summary Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. For scalar parabolic equations for which instability occurs at nonzero wavelength, we show that the associated Ginzburg-Landau equation dominates the dynamics of the nonlinear problem locally, at least over a long timescale. We develop a method which is simpler than previous ones and allows initial conditions of lower regularity. It involves a careful handling of the critical modes in the Fourier-transformed problem and an estimate of Gronwall's type. As an example, we treat the Kuramoto-Shivashinsky equation. Moreover, the method enables us to handle vector-valued problems [see G. Schneider (1992)].     

Uniform estimates for monotone approximation

Uniform estimates are obtained for the monotone approximation of the functions from the generalized Babenko classes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 785–790, June, 1993.     

Optimal error estimates for an approximation of degenerate parabolic problems

A fully discrete scheme for a class of multidimensional degenerate parabolic equations is proposed. The discretization is given by $supoesup piecewise linear finite elements in space and backward differences in time (the smoothing procedure is avoided). Numerical integration is used; hence the proposed method is easy to implement. Optimal error estimates in energy norms are proved for the solutions.     

Error estimates for the numerical approximation of a quaslinear Neumann problem under minimal regularity of the data

The finite element based approximation of a quasilinear elliptic equation of non monotone type with Neumann boundary conditions is studied. Minimal regularity assumptions on the data are imposed. The consideration is restricted to polygonal domains of dimension two and polyhedral domains of dimension three. Finite elements of degree k ≥ 1 are used to approximate the equation. Error estimates are established in the L ²(Ω) and H ¹(Ω) norms for convex and non-convex domains. The issue of uniqueness of a solution to the approximate discrete equation is also addressed.