Negative norm error control for second-kind convolution Volterra equations

Summary. We consider a piecewise constant finite element approximation to the convolution Volterra equation problem of the second kind: find such that in a time interval . An a posteriori estimate of the error measured in the norm is developed and used to provide a time step selection criterion for an adaptive solution algorithm. Numerical examples are given for problems in which is of a form typical in viscoelasticity theory. Received March 5, 1998 / Revised version received November 30, 1998 / Published online December 6, 1999     

Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary. In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law with initial condition . The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is in space-time -norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case. Received October 21, 1999 / Published online February 5, 2001     

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented. Received January 5, 1998 / Revised version received September 7, 1999 / Published online June 8, 2000     

On one approach to a posteriori error estimates for evolution problems solved by the method of lines

Summary. In this paper, we describe a new technique for a posteriori error estimates suitable to parabolic and hyperbolic equations solved by the method of lines. One of our goals is to apply known estimates derived for elliptic problems to evolution equations. We apply the new technique to three distinct problems: a general nonlinear parabolic problem with a strongly monotonic elliptic operator, a linear nonstationary convection-diffusion problem, and a linear second order hyperbolic problem. The error is measured with the aid of the -norm in the space-time cylinder combined with a special time-weighted energy norm. Theory as well as computational results are presented. Received September 2, 1999 / Revised version received March 6, 2000 / Published online March 20, 2001     

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems. Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001     

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix

Summary. Using a slightly different discretization scheme in time and adapting the approach in Nochetto et al. (1998) for analysing the time discretization error in the backward Euler method, we improve on the error bounds derived in (i) Barrett and Blowley (1998) and (ii) Barrett and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix and (i) a logarithmic free energy, and (ii) a non-smooth free energy (the deep quench limit); respectively. Moreover, the improved error bound in the deep quench limit is optimal. Numerical experiments with three components illustrating the above error bounds are also presented. Received June 28, 1999 / Revised version received December 3, 1999 / Published online November 8, 2000     

A posteriori error estimation with the finite element method of lines for a nonlinear parabolic equation in one space dimension

Summary. Convergence of a posteriori error estimates to the true error for the semidiscrete finite element method of lines is shown for a nonlinear parabolic initial-boundary value problem. Received June 15, 1997 / Revised version received May 15, 1998 / Published online: June 29, 1999     

An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy

Summary. An error bound is proved for a fully practical piecewise linear finite element approximation, using a backward Euler time discretization, of the Cahn-Hilliard equation with a logarithmic free energy. Received October 12, 1994     

Spaces of distributions, interpolation by translates of a basis function and error estimates

Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates. Received December 10, 1996 / Revised version received August 29, 1997     

Convergence of Runge–Kutta approximations for parabolic problems with Neumann boundary conditions

Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations, with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded. Received June 25, 1994 / Revised version received February 26, 1996     

Well-posed absorbing layer for hyperbolic problems

Summary. The perfectly matched layer (PML) is an efficient tool to simulate propagation phenomena in free space on unbounded domain. In this paper we consider a new type of absorbing layer for Maxwell's equations and the linearized Euler equations which is also valid for several classes of first order hyperbolic systems. The definition of this layer appears as a slight modification of the PML technique. We show that the associated Cauchy problem is well-posed in suitable spaces. This theory is finally illustrated by some numerical results. It must be underlined that the discretization of this layer leads to a new discretization of the classical PML formulation. Received May 5, 2000 / Published online November 15, 2001     

Asymptotic formula for the error in cardinal interpolation

Summary. We give the asymptotic formula for the error in cardinal interpolation. We generalize the Mazur Orlicz Theorem for periodic function. Received February 22, 1999 / Revised version received October 15, 1999 / Published online March 20, 2001     

Bounds on the error of an approximate invariant subspace for non-self-adjoint matrices

Summary. Suppose one approximates an invariant subspace of an matrix in which in not necessarily self--adjoint. Suppose that one also has an approximation for the corresponding eigenvalues. We consider the question of how good the approximations are. Specifically, we develop bounds on the angle between the approximating subspace and the invariant subspace itself. These bounds are functions of the following three terms: (1) the residual of the approximations; (2) singular--value separation in an associated matrix; and (3) the goodness of the approximations to the eigenvalues. Received December 1, 1992 / Revised version received October 20, 1993     

A posteriori error analysis for numerical approximations of Friedrichs systems

The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments. Received January 10, 1997 / Revised version received March 5, 1998     

An inexactQP-based method for nonlinear complementarity problems

Summary. We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB test problem collection are also reported. Received February 24, 1997 / Revised version received September 5, 1997     

Numerical and theoretical study of a Dual Mesh Method using finite volume schemes for two phase flow problems in porous media

Summary. In this paper we are interested in two phase flow problems in porous media. We use a Dual Mesh Method to discretize this problem with finite volume schemes. In a simplified case (elliptic - hyperbolic system) we prove the convergence of approximate solutions to the exact solutions. We use the Dual Mesh Method in physically complex problems (heterogeneous cases with non constant total mobility). We validate numerically the Dual Mesh Method on practical examples by computing error estimates for different test-cases. Received March 21, 1997 / Revised version received October 13, 1997     

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order , where is the machine precision, the new method computes the eigenvalues to full possible accuracy. Received April 8, 1996 / Revised version received December 20, 1996     

Interior estimates for the wavelet Galerkin method

Summary. In this paper we derive an interior estimate for the Galerkin method with wavelet-type basis. Such an estimate follows from interior Galerkin equations which are common to a class of methods used in the solution of elliptic boundary value problems. We show that the error in an interior domain can be estimated with the best order of accuracy possible, provided the solution is sufficiently regular in a slightly larger domain, and that an estimate of the same order exists for the error in a weaker norm (measuring the effects from outside the domain ). Examples of the application of such an estimate are given for different problems. Received May 17, 1995 / Revised version received April 26, 1996     

Barycentric formulae for cardinal (SINC-) interpolants by Jean-Paul Berrut

Summary. A formula for the efficient evaluation of the (truncated) cardinal series is known to be numerically unstable near the interpolation abscissae. Here it is shown how the series can be evaluated in an entirely stable manner. Received February 14, 2000 / Published online October 16, 2000     

New anisotropic a priori error estimates

Summary. We prove a priori anisotropic estimates for the and interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new error estimate does not require the “maximal angle condition”. The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results. Received March 3, 2000 / Revised version received June 27, 2000 / Published online April 5, 2001