IMA Journal of Numerical Analysis - Advance Access

Smoothness of interpolatory multivariate subdivision in Lie groups

Grohs, P. — 2008-07-31

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.

Convergence rates for adaptive finite elements

Gaspoz, F. D., Morin, P. — 2008-07-30

In this article, we prove that it is possible to construct, using newest vertex bisection, meshes that equidistribute the error in the H¹-norm whenever the function to be approximated can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of partial differential equations. As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite-element methods using Lagrange finite elements of any polynomial degree are obtained.

From high oscillation to rapid approximation III: multivariate expansions

Iserles, A., Norsett, S. P. — 2008-07-25

In this paper, we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace–Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filon-type formulae for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices; hence, we introduce and analyse alternative quadrature schemes that require a relatively modest number of additional function evaluations.

Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Brunner, H., Davies, P. J., Duncan, D. B. — 2008-07-16

Motivated by the problem of developing accurate and stable time-stepping methods for the single-layer potential equation for acoustic scattering from a surface, we present new convergence results for piecewise polynomial discontinuous Galerkin (DG) approximations of a first-kind Volterra integral equation of convolution kernel type, where the kernel K is smooth and satisfies K(0) != 0. We show that an mth degree DG approximation exhibits global convergence of order m when m is odd and order m + 1 when m is even. There is local superconvergence of one order higher (i.e. order m + 1 when m is odd and m + 2 when m is even), but in the even order case, there is superconvergence only if the exact solution u of the equation satisfies u^{(m + 1)}(0) = 0. We also present numerical test results which show that these theoretical convergence rates are optimal.

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

Ferreira, O. P. — 2008-07-08

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.

Compact embeddings of broken Sobolev spaces and applications

Buffa, A., Ortner, C. — 2008-07-04

In this paper, we present several extensions of theoretical tools for the analysis of discontinuous Galerkin (DG) method beyond the linear case. We define broken Sobolev spaces for Sobolev indices in [1, ), and we prove generalizations of many techniques of classical analysis in Sobolev spaces. Our targeted application is the convergence analysis for DG discretizations of energy minimization problems of the calculus of variations. Our main tool in this analysis is a theorem which permits the extraction of a ‘weakly’ converging subsequence of a family of discrete solutions and which shows that any ‘weak limit’ is a Sobolev function. As a second application, we compute the optimal embedding constants in broken Sobolev–Poincaré inequalities.

Interpolation in special orthogonal groups

Shingel, T. — 2008-07-04

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 s(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) —-> s(n) map is multivalued.

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

Dunne, R. K., O'Riordan, E., Shishkin, G. I. — 2008-07-04

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.

Attractors of set-valued partial differential equations under discretization

Kloeden, P. E., Valero, J. — 2008-06-30

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.

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

Elliott, C., Smitheman, S. — 2008-06-25

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.

Nystrom method for systems of integral equations on the real semiaxis

De Bonis, M.C., Mastroianni, G. — 2008-06-20

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.

A recursive -trust-region method for bound-constrained nonlinear optimization

Gratton, S., Mouffe, M., Toint, P. L., Weber-Mendonca, M. — 2008-06-20

A recursive trust-region method is introduced for the solution of bound-constrained nonlinear nonconvex optimization problems for which a hierarchy of descriptions exists. Typical cases are infinite-dimensional problems for which the levels of the hierarchy correspond to discretization levels, from coarse to fine. The new method uses the infinity norm to define the shape of the trust region, which is well adapted to the handling of bounds and also to the use of successive coordinate minimization as a smoothing technique. Numerical tests motivate a theoretical analysis showing convergence to first-order critical points irrespective of the starting point.

Continuous and discrete parabolic operators and their qualitative properties

Farago, I., Horvath, R. — 2008-06-16

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.

Modulated Fourier expansions and heterogeneous multiscale methods

Sanz-Serna, J. M. — 2008-06-10

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.

Discontinuous Galerkin methods for the biharmonic problem

Georgoulis, E. H., Houston, P. — 2008-06-10

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.

An implicit trust-region method on Riemannian manifolds

Baker, C. G., Absil, P.-A., Gallivan, K. A. — 2008-06-05

We propose and analyse an implicit trust-region method in the general setting of Riemannian manifolds. The method is implicit in that the trust region is defined as a superlevel set of the ratio of the actual over-predicted decrease in the objective function. Since this method potentially requires the evaluation of the objective function at each step of the inner iteration, we do not recommend it for problems where the objective function is expensive to evaluate. However, we show that on some instances of a very structured problem—the extreme symmetric eigenvalue problem or equivalently the optimization of the Rayleigh quotient on the unit sphere—the resulting numerical method outperforms state-of-the-art algorithms. Moreover, the new method inherits the detailed convergence analysis of the generic Riemannian trust-region method.

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

Holden, H., Karlsen, KH., Risebro, NH. — 2008-05-26

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation with and . Introducing Riemann invariants R = u_t + cu_x and S = u_t – cu_x, the variational wave equation is equivalent to and with . 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 are nonpositive. The numerical scheme is illustrated on several examples.

Alternate slice-based substructuring in three dimensions

Mihai, L. A., Craig, A. W. — 2008-05-23

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.

Error estimates for Gauss-Turan quadratures and their Kronrod extensions

Milovanovic, G. V., Spalevic, M. M., Pranic, M. S. — 2008-05-16

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).

Stability analysis of general linear methods for the nonautonomous pantograph equation

Huang, C. — 2008-05-13

This paper is concerned with the study of the stability of general linear methods for the nonautonomous pantograph equation. Linear and nonlinear problems are considered separately. We derive the asymptotic stability of numerical methods with strict stability at infinity for neutral equations. Also, we obtain some bounds for the error growth for algebraically stable methods applied to non-neutral equations.

Stochastic variational integrators

Bou-Rabee, N., Owhadi, H. — 2008-05-09

This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds, akin to the Ornstein–Uhlenbeck theory of Brownian motion in a force field. The main result is to derive governing SDEs for such systems from a critical point of a stochastic action. Using this result, the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analogue of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discrete variational principle. The paper shows that the discrete flow of an SVI is almost surely symplectic and in the presence of symmetry almost surely momentum-map preserving. A first-order mean-squared convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid bodies interacting via a potential.

Inf-sup stable finite-element pairs based on dual meshes and bases for nearly incompressible elasticity

Lamichhane, B. P. — 2008-05-09

We consider finite-element methods based on simplices to solve the problem of nearly incompressible elasticity. Two different approaches based, respectively, on dual meshes and dual bases are presented, where in both approaches pressure is discontinuous and can be statically condensed out from the system. These novel approaches lead to displacement-based low-order finite-element methods for nearly incompressible elasticity based on rigorous mathematical framework. Numerical results are provided to demonstrate the efficiency of the approach.

Analysis of a mixed finite-volume discretization of fourth-order equations on general surfaces

Du, Q., Ju, L., Tian, L. — 2008-05-02

In this paper, we study a finite-volume method for the numerical solution of a model fourth-order partial differential equation defined on a smooth surface. The discretization is done via a surface mesh consisting of piecewise planar triangles and its dual surface polygonal tessellation. We provide an error estimate for the approximate solution under the H¹-norm on general regular meshes. Numerical experiments are carried out on various sample surfaces to verify the theoretical results. In addition, when the underlying mesh is constructed by the so-called constrained centroidal Voronoi meshes, we propose a numerically demonstrated superconvergent scheme to compute gradients more accurately.

Trust-region superposition methods for protein alignment

Andreani, R., Martinez, J. M., Martinez, L. — 2008-05-02

Protein alignment is a challenging applied optimization problem. Superposition methods are based on the maximization of a score function with respect to rigid-body modifications of relative positions. The problem of score maximization can be modelled as a continuous nonsmooth optimization problem (low order-value optimization (LOVO)). This allows one to define practical and convergent methods that produce monotone increases of the score. In this paper, trust-region methods are introduced for solving the problem. Numerical results are presented. Computer software related to the LOVO approach for protein alignment is available at www.ime.unicamp.br/ ~ martinez/lovoalign.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Powell, C. E., Elman, H. C. — 2008-04-04

Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

The spectral gradient method for unconstrained optimal control problems

Ardenghi, J. I., Gibelli, T. I., Maciel, M. C. — 2008-04-02

Optimal control problems and their discretized form can be viewed as optimization problems. Kelley and Sachs have already solved the discretized problem by using quasi-Newton methods. In this contribution, the problem is solved by a low-cost algorithm, the spectral gradient method, which is suitable for large-scale problems. The convergence behaviour of the method to finite-dimensional approximation is analysed. Numerical examples are given and the reported results show the good performance of the algorithm when it is applied to large optimal control problems.

From high oscillation to rapid approximation I: modified Fourier expansions

Iserles, A., Norsett, S. P. — 2008-04-02

In this paper, we consider a modification of the classical Fourier expansion, whereby in [– 1, 1] the sin nx functions are replaced by sin (n –1/2)x, n ≥ 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like (n^–2), rather than like (n^–1). We explore theoretical features of these modified Fourier expansions, prove suitable versions of the Fejér and de la Vallée Poussin theorems and expand the coefficients into asymptotic series. This expansion is a key towards the computation of expansion coefficients by asymptotic and Filon-type methods. We explore this issue in some detail and present a number of algorithms which require (m) operations in the computation of the first m expansion coefficients.

A Lions non-overlapping domain decomposition method for domains with an arbitrary interface

Lui, S. H. — 2008-04-02

Lions’ non-overlapping domain decomposition method for the solution of elliptic partial differential equations has been analysed extensively by many authors. There have been numerous works on the convergence of the iterative method as well as variations of it. In the present work, we analyse several formulations of Lions’ method. For two of these, we show that the spectral radius of the operator in the fixed-point iteration for the interface boundary function is bounded above by 1 – O(h^1/2) when the optimal value (O(h^–1/2)) of the parameter in the Robin boundary condition along the artificial interface is used. While this result is already known for rectangular domains with a straight interface, our analysis is valid for essentially arbitrary geometry. The method of Guo and Hou, which is Lions’ method with relaxation, has a simple proof of convergence using our method of analysis. Similarly, we provide a simple proof of convergence of the scheme of Deng.

Linearization of matrix polynomials expressed in polynomial bases

Amiraslani, A., Corless, R. M., Lancaster, P. — 2008-04-01

This paper concerns regular matrix polynomials P() when represented in various polynomial bases (other than the monomials 1, , ², ...). As in the monomial case, matrices of ‘companion’ form play an important part in theory and numerical practice. In particular, they are used here to construct ‘strong linearizations’ of P(). The paper contains three theorems concerning linearizations constructed for representations in a general class of ‘degree-graded’ polynomials, Bernstein polynomials and Lagrange polynomials.

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman, E., Guzman, J., Leykekhman, D. — 2008-03-20

In this paper, we analyse local properties of the continuous interior penalty (CIP) method for a model convection-dominated singularly perturbed convection–diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows thats locally, the CIP method is comparable to the streamline-diffusion or the discontinuous Galerkin methods.

On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Hansen, O., Atkinson, K., Chien, D. — 2008-03-20

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in . We show how hyperinterpolation can be used in connection with the Kumar–Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C^1, (D), > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n^–k), for every if the solution of the nonlinear Poisson equation is in C(D).

A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity

Ern, A., Stephansen, A. F., Zunino, P. — 2008-03-20

We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection–diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Péclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Péclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

Numerically stable LDLT-factorization of -type saddle point matrices

De Niet, A. C., Wubs, F. W. — 2008-03-14

We present a new algorithm that constructs a fill-reducing ordering for a special class of saddle point matrices: the -matrices. This class contains the matrix occurring after discretization of the Stokes equation on a C-grid. The commonly used approach is to construct a fill-reducing ordering for the whole matrix followed by an adaptation of the ordering such that it becomes feasible. We propose to compute first a fill-reducing ordering for an extension of the definite submatrix. This ordering can be easily extended to an ordering for the whole matrix. In this manner, the construction of the ordering is straightforward and it can be computed efficiently. We show that much of the structure of the matrix is preserved during Gaussian elimination. For an -matrix, the preserved structure allows us to prove that any feasible ordering obtained in this way is numerically stable. The growth factor of this factorization is much smaller than the one for general indefinite matrices and is bounded by a number that depends linearly on the number of indefinite nodes. The algorithm allows for generalization to saddle point problems that are not of -type and are nonsymmetric, e.g. the incompressible Navier–Stokes equations (with Coriolis force) on a C-grid. Numerical results for -matrices show that the algorithm is able to produce a factorization with low fill.

Flexible penalty functions for nonlinear constrained optimization: Dedicated to Prof. M. J. D. Powell, pioneer of nonlinear optimization, on occasion of his 70th birthday

Curtis, F. E., Nocedal, J. — 2008-03-14

We propose a globalization strategy for nonlinear constrained optimization. The method employs a ‘flexible’ penalty function to promote convergence, where during each iteration the penalty parameter can be chosen as any number within a prescribed interval, rather than a fixed value. This increased flexibility in the step acceptance procedure is designed to promote long productive steps for fast convergence. An analysis of the global convergence properties of the approach in the context of a line search sequential quadratic programming method and numerical results for the KNITRO software package are presented.

Approximation of the vibration modes of a Timoshenko curved rod of arbitrary geometry

Hernandez, E., Otarola, E., Rodriguez, R., Sanhueza, F. — 2008-03-14

The aim of this paper is to analyse a mixed finite-element method for computing the vibration modes of a Timoshenko curved rod with arbitrary geometry. Optimal order error estimates are proved for displacements, rotations and shear stresses of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are essentially independent of the thickness of the rod, which leads to the conclusion that the method is locking-free. Numerical tests are reported in order to assess the performance of the method.

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

Hackbusch, W., Kress, W., Sauter, S. A. — 2008-03-11

Many important physical applications are governed by the wave equation. The formulation as time domain boundary integral equations involves retarded potentials. For the numerical solution of this problem, we employ the convolution quadrature method for the discretization in time and the Galerkin boundary element method for the space discretization. We introduce a simple a priori cut-off strategy where small entries of the system matrices are replaced by zero. The threshold for the cut-off is determined by an a priori analysis which will be developed in this paper. This analysis will also allow to estimate the effect of additional perturbations such as panel clustering and numerical integration on the overall discretization error. This method reduces the storage complexity for time domain integral equations from O(M²N) to O(M² N^1/2 log M), where N denotes the number of time steps and M is the dimension of the boundary element space.

Developments of NEWUOA for minimization without derivatives

Powell, M. J. D. — 2008-02-27

The NEWUOA software is described briefly, with some numerical results that show good efficiency and accuracy in the unconstrained minimization without derivatives of functions of up to 320 variables. Some preliminary work on an extension of NEWUOA that allows simple bounds on the variables is also described. It suggests a variation of a technique in NEWUOA for maintaining adequate linear independence in the interpolation conditions that are used, which leads to five versions of the technique including the original one. Numerical experiments suggest that the new versions have some merit, but the details of the calculations are influenced strongly by computer rounding errors. The dependence of the number of iterations on the number of interpolation conditions is also investigated numerically. A surprising case with n = 160 is found, n being the number of variables, where the number of iterations is reduced when the number of conditions is decreased from 2n + 1 to n + 6. The given conclusions may assist the development of some new software for unconstrained optimization.

A safeguarded dual weighted residual method

Nochetto, R. H., Veeser, A., Verani, M. — 2008-02-27

The dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case, its performance is generally superior than that of global ‘energy norm’ error estimators which are ‘unconditionally’ reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example.

Layer-adapted meshes for a linear system of coupled singularly perturbed reaction-diffusion problems

Linss, T., Madden, N. — 2008-02-16

We consider a system of ≥ 2 one-dimensional singularly perturbed reaction–diffusion equations coupled at the zero-order term. The second derivative of each equation is multiplied by a distinct small parameter. We show how to decompose the solution to the problem into regular and layer parts. Properties of the discretized operator are established using discrete Green's functions. We prove that a central difference scheme on certain layer-adapted meshes converges independently of the perturbation parameters. Supporting numerical examples confirm our theoretical results.

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

Gatica, G. N., Meddahi, S., Oyarzua, R. — 2008-02-16

We consider a porous medium entirely enclosed within a fluid region and present a well-posed conforming mixed finite-element method for the corresponding coupled problem. The interface conditions refer to mass conservation, balance of normal forces and the Beavers–Joseph–Saffman law, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite-element subspaces defining the discrete formulation employ Bernardi–Raugel and Raviart–Thomas elements for the velocities, piecewise constants for the pressures and continuous piecewise-linear elements for the Lagrange multiplier. We show stability, convergence and a priori error estimates for the associated Galerkin scheme. Finally, we provide several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence.

A new approach to energy-based sparse finite-element spaces

Todor, R.-A. — 2008-02-16

We show that the logarithmic factor in the standard error estimate for sparse finite element (FE) spaces in arbitrary dimension d is removable in the energy (H¹) norm. Via a penalized sparse grid condition, we then propose and analyse a new version of the energy-based sparse FE spaces introduced first in Bungartz (1992, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation. Munich, Germany: TU München) and known to satisfy an optimal approximation property in the energy norm.

Dynamic frictionless contact in linear viscoelasticity

Ahn, J., Stewart, D. E. — 2008-02-16

In this work, we formulate a dynamic frictionless contact problem with linear viscoelasticity of Kelvin–Voigt type, based on the Signorini contact conditions. We show existence of solutions, and investigate the possibility for obtaining an energy balance. Employing time discretization and the finite-element method, we compute numerical solutions. Our numerical scheme is implemented with non-smooth Newton's method which solves the complementarity problem. The numerical results support the idea that the energy losses in the limit of the numerical solution are equal to the losses due to viscosity.

Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

Nielsen, B. F., Tveito, A., Hackbusch, W. — 2008-02-06

We study the eigenvalues of the operator generated by using the inverse of the Laplacian as a preconditioner for self-adjoint second-order elliptic partial differential equations with smooth coefficients. It is well-known that the spectral condition number of the preconditioned operator can be bounded by , where k is the uniformly positive coefficient of the second-order elliptic equation. The purpose of this paper is to study the spectrum of the preconditioned operator. We will show that there is a strong relation between the spectrum of this operator and the range of the coefficient function. In the continuous case, we prove, both for mappings defined on Sobolev spaces and in terms of generalized functions, that the spectrum of the preconditioned operator contains the range of the coefficient function k. In the discrete case, we indicate by numerical examples that the entire discrete spectrum is approximately given by values of k.

Recurrence and asymptotics for orthonormal rational functions on an interval

Deckers, K., Bultheel, A. — 2008-01-29

Let µ be a positive bounded Borel measure on a subset I of the real line and A = {₁, ..., _n} a sequence of arbitrary ‘complex’ poles outside I. Suppose {₁, ..., _n} is the sequence of rational functions with poles in A orthonormal on I with respect to µ. First, we are concerned with reducing the number of different coefficients in the three-term recurrence relation satisfied by these orthonormal rational functions. Next, we consider the case in which I = [– 1, 1] and µ satisfies the Erdos–Turán condition µ' > 0 a.e. on I (where µ' is the Radon–Nikodym derivative of the measure µ with respect to the Lebesgue measure) to discuss the convergence of _n₊₁(x)/_n(x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation. Finally, we give a strong convergence result for _n(x) under the more restrictive condition that µ satisfies the Szego condition (1 – x²)^–1/2 log µ'(x) L¹([ – 1, 1]).

Multigrid methods with Powell Sabin splines: Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday and in honour of his many contributions to numerical analysis

Speleers, H., Dierckx, P., Vandewalle, S. — 2007-11-27

This paper presents a multigrid algorithm for the solution of the linear systems that arise from a finite-element discretization of second-order elliptic partial differential equations with Powell–Sabin (PS) splines. We show that the method yields a uniform convergence in the l₂-norm which is independent of the mesh size. We also briefly consider the use of PS splines for the fourth-order biharmonic problem.

Interpolation and scattered data fitting on manifolds using projected Powell Sabin splines: Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday

Davydov, O., Schumaker, L. L. — 2007-10-26

We present methods either for interpolating data or for fitting scattered data on a 2D smooth manifold . The methods are based on a local bivariate Powell–Sabin interpolation scheme, and make use of a family of charts {(U, )} satisfying certain conditions of smooth dependence on . If is a C²-manifold embedded into , then projections into tangent planes can be employed. The data-fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function.