IMA Journal of Numerical Analysis - current issue

The QR algorithm: 50 years later its genesis by John Francis and Vera Kublanovskaya and subsequent developments

Golub, G., Uhlig, F. — 2009-07-02

Fifty years after the invention of the QR algorithm by John Francis and Vera Kublanovskaya we reconstruct the ideas and the influences that led to its genesis from the originators’ own recollections and their sources and give an account of some of its subsequent developments.

Error estimates for Gauss-Turan quadratures and their Kronrod extensions

Milovanovic, G. V., Spalevic, M. M., Pranic, M. S. — 2009-07-02

We study the kernel K_n_{, s}(z) of the remainder term R_n_{, s}(f) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod, using the modulus of the difference of Gauss–Turán quadratures and their Kronrod extensions, we derive new error estimates for Gauss–Turán quadratures and compare them with the effective L¹-error bounds derived in Milovanovic & Spalevic (2005, BIT, 45, 117–136).

Alternate slice-based substructuring in three dimensions

Mihai, L. A., Craig, A. W. — 2009-07-02

The slice-based substructuring methods introduced in this paper are Schur complement solvers for the decomposition of a three-dimensional domain into multiple disjoint subdomains with interior crosspoints. The subdomains are assembled into nonoverlapping slices such that the edges of each slice lie on the boundary of the given domain and the union of the faces between slices contains all of the interior vertices. For the subproblems corresponding to the various faces, a direct fast Poisson solver is used. Scalability is achieved in two stages where the slices change such that the faces between slices at one stage are orthogonal to the faces between slices at the other. The two stages guarantee a good rate of convergence of the resulting preconditioned iterative procedure, which is optimal with respect to the partitioning parameters.

A convergent finite-difference method for a nonlinear variational wave equation

Holden, H, Karlsen, K. H., Risebro, N. H. — 2009-07-02

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation u_tt – c(u)(c(u)u_x)_x = 0 with u|_t₌₀ = u₀ and u_t|_t=0 = v₀. Introducing Riemann invariants R = u_t + cu_x and S = u_t – cu_x, the variational wave equation is equivalent to R_t – cR_x (R² – S²) and S_t + cS_x = – (R² – S²) with = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R|_t₌₀, S|_t₌₀ L¹(R) L³(R) are nonpositive. The numerical scheme is illustrated on several examples.

Discontinuous Galerkin methods for the biharmonic problem

Georgoulis, E. H., Houston, P. — 2009-07-02

This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems involving the biharmonic operator. The first part extends the unified approach of Arnold et al. (2001/2002, SIAM J. Numer. Anal., 39, 1749-–1779) developed for the Poisson problem, to the design of DG methods via an appropriate choice of numerical flux functions for fourth-order problems; as an example, we retrieve the interior penalty DG method developed by Süli & Mozolevski (2007, Comput. Methods Appl. Mech. Eng., 196, 1851-–1863). The second part of this work is concerned with a new a priori error analysis of the hp-version interior penalty DG method, when the error is measured in terms of both the energy norm and the L²-norm, as well as certain linear functionals of the solution, for elemental polynomial degrees p ≥ 2. Also, provided that the solution is piecewise analytic in an open neighbourhood of each element, exponential convergence is also proved for the p-version of the DG method. The sharpness of the theoretical developments is illustrated by numerical experiments.

Modulated Fourier expansions and heterogeneous multiscale methods

Sanz-Serna, J. M. — 2009-07-02

We show that, for highly oscillatory ordinary differential equation problems, the modulated Fourier expansion approach can be advantageously used to understand and analyse the heterogeneous multiscale methods introduced by E, Engquist and their co-workers.

Continuous and discrete parabolic operators and their qualitative properties

Farago, I., Horvath, R. — 2009-07-02

The basic requirement of numerical methods is convergence. However, from the practical point of view, it is generally not sufficient to construct convergent numerical methods for the solutions of partial differential equations. The qualitative adequateness of the methods is also an issue. The numerical solutions should mirror the characteristic properties of the original physical process that is modelled by the differential equation. In this paper, we give three important qualitative properties of parabolic partial differential equations: the maximum–minimum principle and its different versions, the non-negativity preservation and the maximum norm contractivity. The investigation of these properties is motivated by different physical principles. We formulate the analogues of the properties for general discrete operators and we analyse the conditions and the relations between the properties for both the continuous and the discrete operators. The approximation properties of the discrete operators are also analysed. The results of the paper are applied to the finite-difference solution methods of parabolic initial boundary-value problems.

Nystrom method for systems of integral equations on the real semiaxis

De Bonis, M. C., Mastroianni, G. — 2009-07-02

In this paper, the authors introduce a Nyström method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schrödinger equation.

Numerical analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture

Elliott, C. M., Smitheman, S. A. — 2009-07-02

The Osher–Solé–Vese (OSV) model, which is the gradient flow of an energy consisting of the total variation functional plus an H^–1 fidelity term, is studied. In this paper, we build on the analysis of the OSV model which we gave in Elliott & Smitheman (2007, Comm. Pure Appl. Anal., in press). We introduce backward Euler finite-element approximations to a regularized version of the OSV initial boundary-value problem (IBVP) and to a weak formulation of the original problem. Well-posedness and unconditional Lyapunov stability of these fully discrete schemes are proved. Convergence results as the spatial mesh parameter, the time step size and the regularization parameter tend to 0 are proved. Rates of convergence as the time step size and the regularization parameter tend to 0 are found. The existence, uniqueness and Lyapunov stability of a solution to a linearly implicit finite-element approximation to the regularized version of the OSV IBVP are also proved.

Attractors of set-valued partial differential equations under discretization

Kloeden, P. E., Valero, J. — 2009-07-02

The approximation of the global attractor of a dissipative set-valued reaction–diffusion equation is investigated when a Galerkin approximation is used to obtain a finite-dimensional inclusion equation, to which the linear implicit Euler scheme is then applied. The existence and upper semicontinuous convergence of the various attractors with decreasing time step and increasing dimension are established. The equivalence of the attractors with those of the corresponding convexified systems is also shown.

Fitted mesh numerical methods for singularly perturbed elliptic problems with mixed derivatives

Dunne, R. K., O'Riordan, E., Shishkin, G. I. — 2009-07-02

A class of singularly perturbed convection–diffusion problems is considered which contain a mixed derivative term. We consider the case when exponential boundary layers are present in the solutions of problems from this class. Under appropriate assumptions on the data of the problem, we construct a decomposition of the solution into regular and layer components. We then introduce a numerical method on a piecewise-uniform fitted mesh. Excluding a neighbourhood of one of the corners, it is shown that in the perturbed case (i.e. when the perturbation parameter is small relative to the inverse of the number of mesh intervals in both coordinate directions), the approximations generated by the method converge uniformly with respect to the singular perturbation parameter. Finally, numerical examples are presented that illustrate the theoretical result.

Interpolation in special orthogonal groups

Shingel, T. — 2009-07-02

In this paper, we propose a scheme to generate interpolating curves in Lie groups, focusing on a special orthogonal group SO(n) due to its practical importance. Our technique is based on the exponential representation of the elements of the group, which allows as to transfer the problem to the corresponding Lie algebra (n) and benefit from the linearity of this space. Due to the exponential representation, we can obtain a high degree of smoothness of an interpolating curve. The underlying problem is challenging because the standard SO(n) -> (n) map is multivalued.

Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle

Ferreira, O. P. — 2009-07-02

A local convergence analysis of Newton's method for solving nonlinear equations, based on Kantorovich's majorant principle, is presented in this paper. This analysis provides a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allows us to obtain the optimal convergence radius, the biggest range for the uniqueness of the solution, and to unify some previous and unrelated results.

Smoothness of interpolatory multivariate subdivision in Lie groups

Grohs, P. — 2009-07-02

Nonlinear subdivision schemes that operate on manifolds are of use whenever manifold-valued data have to be processed in a multiscale fashion. This paper considers the case where the manifold is a Lie group and the nonlinear subdivision schemes are derived from linear interpolatory ones by the so-called log–exp analogy. The main result of the paper is that a multivariate interpolatory Lie-group-valued subdivision scheme derived from a linear scheme is at least as smooth as the linear scheme, where smoothness is understood in terms of Hölder exponents.

A symmetric collocation method with fast evaluation

Johnson, M. J. — 2009-07-02

Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function interpolation. As with radial basis function interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating-point operations, of evaluating the obtained function. When solving a linear partial differential equation, one usually has some freedom in choosing the collocation points. We explain how this freedom can be exploited to allow the fast evaluation of the obtained function provided the basic function is chosen as a tensor product of compactly supported piecewise polynomials. Our proposed fast evaluation method, which is exact in exact arithmetic, is initially designed and analysed in the univariate case. The multivariate case is then reduced, recursively, to multiple univariate evaluations. Along with the theoretical development of the method, we report the results of selected numerical experiments which help to clarify expectations.

The LBB condition in fractional Sobolev spaces and applications

Guermond, J.-L. — 2009-07-02

The present work focuses on the approximation of the stationary Stokes equations by means of finite-element-like Galerkin methods. It is shown that, provided the velocity space and the pressure space are compatible in some sense, a Ladyzhenskaya–Babuska–Brezzi condition holds in the fractional Sobolev spaces H^s(), s [0, 1]. This result is illustrated in two applications.

Remarks on the implementation of the fast marching method

Rasch, C., Satzger, T. — 2009-07-02

The fast marching algorithm computes an approximate solution to the eikonal equation in time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv et al. (2006 J. Comput. Phys., 212, 393–399) have suggested using an untidy priority queue, reducing the overall complexity to at the price of a small error in the computed solution. In this paper we give an explicit estimate of the error introduced, which is based on a discrete comparison principle. This estimate implies, in particular, that the choice of an accuracy level that is independent of the speed function F results in the complexity bound being . A numerical experiment illustrates this robustness problem for large ratios F max /F min.

A derivative-free nonmonotone line search and its application to the spectral residual method

Cheng, W., Li, D.-H. — 2009-07-02

In this paper we propose a derivative-free nonmonotone line search for solving large-scale nonlinear systems of equations. Under appropriate conditions, we show that the spectral residual method with this line search is globally convergent. We also present some numerical experiments. The results show that the spectral residual method with the new nonmonotone line search is promising.