IMA Journal of Numerical Analysis - Advance Access

Numerical approximations to the mass transfer problem on compact spaces

Gabriel, J. R., Gonzalez-Hernandez, J., Lopez-Martinez, R. R. — 2009-07-02

This paper presents a numerical approximation for the value of the Monge–Kantorovich mass transfer (MT) problem on compact metric spaces. A sequence of transportation problems is built and it is proven that the value of the MT problem is the limit of the optimal values of these problems. Moreover, we give an error bound for the numerical approximation. A couple of illustrative computational examples are presented.

On convergence of a fitted finite-volume method for the valuation of options on assets with stochastic volatilities

Huang, C.-S., Hung, C.-H., Wang, S. — 2009-07-01

In this paper we present a convergence analysis for a fitted finite-volume discretization method for the two-dimensional Black–Scholes equation arising in the Hull–White model for pricing European options with stochastic volatility. We first formulate the finite-volume method as a nonconforming Petrov–Galerkin finite–element method with each basis function of the trial space being determined by a set of two-point boundary-value problems defined on element edges. We then show that the bilinear form of the finite-element method is coercive and continuous. Finally we establish an upper bound of order (h) on the discretization error of method, where h denotes the mesh parameter of the discretization.

Nonoverlapping Schwarz algorithm for solving two-dimensional m-DDFV schemes

Boyer, F., Hubert, F., Krell, S. — 2009-06-30

We propose a nonoverlapping Schwarz algorithm for solving ‘discrete duality finite-volume’ (DDFV) schemes on general meshes. In order to handle this problem the first step is to propose and study a convenient DDFV scheme for anisotropic elliptic problems with mixed Dirichlet/Fourier boundary conditions. Then we are able to build the corresponding Schwarz algorithm and to prove its convergence to the solution of the DDFV scheme on the initial domain. We finally give some numerical results in the case where the Schwarz iterations are used both as a solver and as a preconditioner.

The convection-diffusion Petrov-Galerkin story

Morton, K. W. — 2009-06-23

The term ‘Petrov–Galerkin method’ is probably due to Ron Mitchell and his collaborators, and he was certainly very active in studying the application of finite-element methods to second-order partial differential equations ‘with significant first derivatives’. Our aim in the present paper is to trace links between such early methods and the more recent discontinuous Galerkin methods—not only the methods but also their analysis. Also as Mitchell, like the author, was initially steeped in finite-difference methods, we shall sometimes use their manipulation in our analysis.

A survey of results on the q-Bernstein polynomials

Phillips, G. M. — 2009-06-23

It is now nearly a century since S. N. Bernstein introduced his well-known polynomials. This paper is concerned with generalizations of the Bernstein polynomials, mainly with the so called q-Bernstein polynomials. These are due to the author of this paper and are based on the q integers. They reduce to the Bernstein polynomials when we put q = 1 and share the shape-preserving properties of the Bernstein polynomials when q (0, 1). This paper also describes another earlier generalization of the Bernstein polynomials, a sequence of rational functions that are also based on the q-integers, proposed by A. Lupa, and two even earlier generalizations due to D. D. Stancu. The present author summarizes various results, due to a number of authors, that are concerned with the q-Bernstein polynomials and with Stancu's two generalizations.

Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation

McLean, W., Thomee, V. — 2009-06-19

In a previous paper, McLean & Thomeé(2009, J. Integr. Equ. Appl. (to appear)), we studied three numerical methods for the discretization in time of a fractional-order evolution equation in a Banach space framework. Each of the methods applied a quadrature rule to a contour integral representation of the solution in the complex plane, where for each quadrature point an elliptic boundary-value problem had to be solved to determine the value of the integrand. The first two methods involved the Laplace transform of the forcing term, but the third did not. We analysed both the quadrature error and the error arising from a spatial discretization by finite elements, measured in the L₂-norm. The present work extends our earlier results by proving error bounds in the technically more complicated case of the maximum norm. We also establish new regularity properties for the exact solution that are needed for our analysis.

A restarted Lanczos approximation to functions of a symmetric matrix

Ilic, M., Turner, I. W., Simpson, D. P. — 2009-06-17

In this paper the error term in the ‘first’ method of Eiermann & Ernst(2006, SIAM J. Numer. Anal., 44, 2481–2504) for restarting the Lanczos approximation of the matrix–vector product f(A)b, where A Rⁿ ^{x n} is symmetric, is re-derived and expressed as an explicit partial fraction expansion of the divided differences. The partial fraction representation makes the new variant slightly more stable (albeit still unstable) than the former method because it requires fewer finite-difference evaluations. We then present an error bound for the restarted Lanczos approximation of f(A)b for symmetric positive-definite A when f is in a particular class of completely monotone functions and illustrate for some important matrix function applications the usefulness of these bounds for terminating the restart process once the desired accuracy in the matrix function approximation has been achieved. Finally, in an attempt to overcome the inherent instability of our restart procedure, we propose a simple heuristic that identifies when to halt the iterations.

Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces

Eymard, R., Gallouet, T., Herbin, R. — 2009-06-16

A symmetric discretization scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces that may, for instance, be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate for several numerical examples. The convergence of the approximate solution to the continuous solution is proved for general (possibly discontinuous) tensors and general (possibly nonconforming) meshes and with no regularity assumption on the solution. An error estimate is then deduced under suitable regularity assumptions on the solution.

A preconditioned Newton algorithm for the nearest correlation matrix

Borsdorf, R., Higham, N. J. — 2009-06-08

Various methods have been developed for computing the correlation matrix nearest in the Frobenius norm to a given matrix. We focus on a quadratically convergent Newton algorithm recently derived by Qi and Sun. Various improvements to the efficiency and reliability of the algorithm are introduced. Several of these relate to the linear algebra: the Newton equations are solved by minres instead of the conjugate gradient method, as it more quickly satisfies the inexact Newton condition; we apply a Jacobi preconditioner, which can be computed efficiently even though the coefficient matrix is not explicitly available; an efficient choice of eigensolver is identified; and a final scaling step is introduced to ensure that the returned matrix has unit diagonal. Potential difficulties caused by rounding errors in the Armijo line search are avoided by altering the step selection strategy. These and other improvements lead to a significant speed-up over the original algorithm and allow the solution of problems of dimension a few thousand in a few tens of minutes.

A priori and a posteriori error analyses of an augmented discontinuous Galerkin formulation

Barrios, T. P., Bustinza, R. — 2009-06-08

We use Galerkin least squares terms to develop a more general stabilized discontinuous Galerkin method for elliptic problems in the plane with mixed boundary conditions. The unique solvability and optimal rate of convergence of this scheme, with respect to the h-version, are established. Furthermore, we include the corresponding a posteriori error analysis, which results in a reliable and efficient estimator. Finally, we present several numerical examples that show the capability of the adaptive algorithm to localize the singularities, confirming the theoretical properties of the a posteriori error estimate.

Comparisons between pseudospectral and radial basis function derivative approximations

Fornberg, B., Flyer, N., Russell, J. M. — 2009-06-02

Fourier-based pseudospectral (PS) methods have been used since the 1970s for obtaining spectrally accurate solutions to partial differential equations (PDEs) in periodic geometries. Radial basis functions (RBFs) were introduced about the same time for interpolation on scattered nodes in irregular geometries. As was later recognized, they can also be used for accurate numerical solution of PDEs. Although the main strength of RBFs lies in their outstanding geometric flexibility, also offering possibilities of spectral accuracy over irregularly shaped finite domains, it is still of interest to compare them against Fourier-based PS methods in the extremely simple geometries (infinite or periodic domains) where the latter can also be used. Mostly by means of heuristic arguments and graphical illustrations based on Fourier analysis and numerical experiments, we show that there are notable differences (more pronounced in increasing numbers of dimensions) in how the two spectral approaches approximate derivatives.

Trees, B-series and exponential integrators

Butcher, J. C. — 2009-05-19

Questions concerning the accuracy of numerical methods for differential equations are often analysed using B-series and other formulations based on rooted trees. The analysis of numerical methods, such as Rosenbrock and certain exponential methods, requires an additional algebraic structure to represent the direct use of Jacobian matrices in the computation. It is shown how this can be done within a context containing a broad review of the existing B-series formulation.

Dimension splitting for quasilinear parabolic equations

Hansen, E., Ostermann, A. — 2009-05-19

In the current paper we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman–Rachford splittings. The analysis is, in particular, applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissipative operators, and we give both a summary of the used theory and some extensions of the classical results. The derived convergence results are illustrated by numerical experiments.

Improved contour integral methods for parabolic PDEs

Weideman, J. A. C. — 2009-04-17

One way of computing the matrix exponential that arises in semidiscrete parabolic partial differential equations is via the Dunford–Cauchy integral formula. The integral is approximated by the trapezoidal or midpoint rules on a Hankel contour defined by a suitable change of variables. In a recent paper by Weideman & Trefethen (2007, Math. Comput., 76, 1341–1356) two widely used contours were analysed. Estimates for the optimal parameters that define these contours were proposed. In this paper this analysis is extended in two directions. First, the effect of roundoff error is now included in the error model. Second, we extend the results to the case of a model convection–diffusion equation, where a large convective term causes the matrix to be highly non-normal.

Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems

Burman, E., Hansbo, P. — 2009-04-07

In this paper we propose a class of jump-stabilized Lagrange multiplier methods for the finite-element solution of multidomain elliptic partial differential equations using piecewise-constant or continuous piecewise-linear approximations of the multipliers. For the purpose of stabilization we use the jumps in derivatives of the multipliers or, for piecewise constants, the jump in the multipliers themselves, across element borders. The ideas are illustrated using Poisson's equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

Robust estimates for the approximation of the dynamic consolidation problem

Sauter, M., Wieners, C. — 2009-03-30

We consider stable discretizations in time and space for the linear dynamic consolidation problem describing wave propagation in a porous solid skeleton that is fully saturated with an incompressible fluid. Introducing the hydrostatic pressure, the flow problem is described by Darcy's law. In particular, we discuss the case of nearly-impermeable solids, which requires inf–sup stable discretizations in space for the limiting saddle point problem. Together with an (implicit) Newmark discretization in time, we derive convergence estimates for the fully-discrete scheme that are robust with respect to the coupling parameter of fluid and solid.

A semi-Lagrangian-Galerkin projection scheme for convection equations

Bermejo, R., Carpio, J. — 2009-03-30

We introduce in this paper a semi-Lagrangian–Galerkin projection scheme to discretize backwards in time along the characteristics the convection terms of convection–diffusion equations. The scheme consists of a transport step in which the elements of the fixed mesh are transported backwards along the characteristic curves, thus generating a new mesh composed of curved elements, followed by an approximate L²-projection onto the finite-element space associated with the transported mesh. The new scheme is to some extent related to the so-called Lagrange–Galerkin (or characteristic-Galerkin) methods, but it may be more efficient because the number of trajectories per element to be calculated in the new scheme is smaller than that of the conventional characteristic-Galerkin scheme. It is also proved that, for linear convection problems with the velocity sufficiently smooth, the new scheme is unconditionally stable in the L²-norm and its order of convergence is , where m is the degree of the polynomials of the finite-element space, and the velocity is in L(0, T;W^q^+1,) with integer q≥ 1.

Efficient numerical solution of the one-dimensional Schrodinger eigenvalue problem using Magnus integrators

Ledoux, V., Van Daele, M., Vanden Berghe, G. — 2009-03-27

We discuss a new numerical method, based on a modified Magnus integrator, to solve the Sturm–Liouville eigenvalue problem in its Schrödinger form. A modified Magnus integrator has already been used by Degani & Schiff (2006, J. Comput. Appl. Math., 193, 413–436) to approximate the oscillating solution of a Schrödinger problem over the classically allowed region. Here we show that the technique can be successfully extended to the nonoscillatory classically forbidden region. This means that the modified Magnus integrator can be used to propagate the solution over the whole integration interval and is well suited for application in a shooting process to locate the eigenvalues. Such a shooting procedure is formulated and shown to allow the efficient computation of a range of eigenvalues and eigenfunctions.

An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem

Brenner, S. C., Gudi, T., Sung, L.-y. — 2009-03-27

A reliable and efficient residual-based a posteriori error estimator is derived for a quadratic C⁰-interior penalty method for the biharmonic problem on polygonal domains. The performance of the estimator is illustrated by numerical experiments.

Smoothness equivalence properties of interpolatory Lie group subdivision schemes

Xie, G., Yu, T. P. -Y. — 2009-03-25

We prove that any interpolatory Lie group subdivision scheme based on combining a linear interpolatory subdivision scheme with the log–exp adaption to Lie-group-valued data in Ur Rahman et al. (2005, Multiscale Model. Simul., 4, 1201–1232) produces parameterized curves on the Lie group that are as smooth as the smoothness of —no matter how smooth is. We present both an extrinsic proof and an intrinsic proof. We discuss two variations of our main result. (i) We illustrate how smoothness equivalence can break down in a variant of the original log– exp scheme. (ii) We show that the main result of this paper can be easily extended to a multivariate setting.

Finite-element approximation of non-Fickian polymer diffusion

Bauermeister, N., Shaw, S. — 2009-03-16

The problem of nonlinear non-Fickian polymer diffusion as modelled by a diffusion equation with an adjoined spatially local evolution equation for a viscoelastic stress is considered. We present numerical schemes based, spatially, on the Galerkin finite-element method and, temporally, on the Crank–Nicolson method. Special attention is paid to linearizing the discrete equations by extrapolating the value of the nonlinear term from previous time steps. Optimal a priori error estimates are given, based on the assumption that the exact solution possesses certain regularity properties, and numerical experiments are given to support these error estimates.

Numerical estimation of the Robin coefficient in a stationary diffusion equation

Jin, B., Zou, J. — 2009-03-16

A finite-element method is proposed for the nonlinear inverse problem of estimating the Robin coefficient in a stationary diffusion equation from boundary measurements of the solution and the heat flux. The inverse problem is formulated as an output least squares optimization problem with an appropriate regularization, then the finite-element method is employed to discretize the nonlinear optimization system. Mathematical properties of both the continuous and the discrete optimization problems are investigated. The conjugate gradient method is employed to solve the optimization problem, and an efficient preconditioner via the Sobolev inner product is also suggested. Numerical results for several two-dimensional problems are presented to illustrate the efficiency of the proposed algorithm.

Discretization error and modelling error in the context of the rapid inflation of hyperelastic membranes

Shaw, S., Warby, M. K., Whiteman, J. R. — 2009-03-13

The computational modelling of the rapid large inflation of hyperelastic circular sheets modelled as axisymmetric membranes is treated, with the aim of estimating engineering quantities of interest and their errors. Fine (involving inertia terms) and coarse (quasi-static) models of the inflation are considered and, using goal-oriented techniques, both modelling and discretization errors are estimated. Numerical results involving only discretization errors for the quasi-static problem and both modelling and discretization errors for the dynamic problem are presented.

Numerical analysis of a finite-element method for the axisymmetric eddy current model of an induction furnace

Bermudez, A., Reales, C., Rodriguez, R., Salgado, P. — 2009-03-12

The aim of this paper is to analyse a finite-element method to solve an eddy current problem arising from the modelling of an induction furnace. By taking advantage of the cylindrical symmetry, the three-dimensional problem reduces to a two-dimensional one on a meridional section, provided that the current density, written in cylindrical coordinates, has only an azimuthal component. A mixed formulation in appropriate weighted Sobolev spaces is given. The existence and uniqueness of the solution are proved by analysing an equivalent weak formulation. Moreover, an additional regularity result is proved under suitable assumptions on the physical coefficients. The problem is discretized by standard finite elements and a priori error estimates are proved. Finally, some numerical experiments that allow an assessment of the performance of the method are reported.

A mesh-free partition of unity method for diffusion equations on complex domains

Eigel, M., George, E., Kirkilionis, M. — 2009-03-06

We present a numerical method for solving partial differential equations on domains with distinctive complicated geometrical properties. These will be called complex domains. Such domains occur in many real-world applications, for example in geology or engineering. We are, however, particularly interested in applications stemming from the life sciences, especially cell biology. In this area complex domains, such as those retrieved from microscopy images at different scales, are the norm and not the exception. Therefore geometry is expected to directly influence the physiological function of different systems, for example signalling pathways. New numerical methods that are able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restrictions in the past. The approximation approach presented here for such problems is based on a promising mesh-free Galerkin method: the partition of unity method (PUM). We introduce the main approximation features and then focus on the construction of appropriate covers as the basis of discretizations. As a main result we present an extended version of cover construction, ensuring fast convergence rates in the solution process. Parametric patches are introduced as a possible way of approximating complicated boundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergence behaviour of the PUM are demonstrated in several numerical examples.

An h-narrow band finite-element method for elliptic equations on implicit surfaces

Deckelnick, K., Dziuk, G., Elliott, C. M., Heine, C.-J. — 2009-03-04

In this article we define a finite-element method for elliptic partial differential equations (PDEs) on curves or surfaces, which are given implicitly by some level set function. The method is specially designed for complicated surfaces. The key idea is to solve the PDE on a narrow band around the surface. The width of the band is proportional to the grid size. We use finite-element spaces that are unfitted to the narrow band, so that elements are cut off. The implementation nevertheless is easy. We prove error estimates of optimal order for a Poisson equation on a surface and provide numerical tests and examples.

Natural p-BEM for the electric field integral equation on screens

Bespalov, A., Heuer, N. — 2009-02-27

In this paper we analyse the p-version of the boundary element method for the electric field integral equation on a plane open surface with polygonal boundary. We prove the convergence of the p-version with Raviart–Thomas parallelogram elements and derive an a priori error estimate that takes into account the strong singular behaviour of the solution at the edges and corners of the surface. The key ingredient of our analysis is the orthogonality of discrete Helmholtz decompositions in a Sobolev space of order .

Weak convergence in the Prokhorov metric of methods for stochastic differential equations

Charbonneau, B., Svyrydov, Y., Tupper, P. F. — 2009-02-27

We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen–Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions, assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding.

A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel

Mustapha, K., Mustapha, H. — 2009-02-26

We study a generalized extrapolated Crank–Nicolson scheme for the time discretization of a semilinear integro-differential equation with a weakly singular kernel, in combination with a space discretization by linear finite elements. The scheme uses variable grids 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 k² + h², where k and h are the parameters for the time and space meshes, respectively. The results of numerical computations demonstrate the convergence of our scheme.

A note on radial basis function interpolant limits

Buhmann, M. D., Dinew, S., Larsson, E. — 2009-02-26

Radial basis functions (RBFs) are very useful in multivariate interpolation because of their ability to produce highly accurate results for scattered data. Many of them, especially the Gaussian RBF and the multiquadric RBF, contain parameters that need to be adjusted in order to improve the approximations. In fact, it is often of interest to let the parameters tend to certain limits. Here we study if and when the limits of RBF interpolants with parameters exist. Mainly, the dependence of the limit on the properties of the radial functions and on the geometries of the data points is investigated, and some examples are provided.

Nonautonomous stability of linear multistep methods

Boutelje, B. R., Hill, A. T. — 2009-02-20

A linear scalar nonautonomous initial-value problem (IVP) is governed by a scalar (t) with a nonpositive real part. For a wide class of linear multistep methods, including BDF4–6, it is shown that negative real (t) may be chosen to generate instability in the method when applied to the IVP. However, a uniform-in-time stability result holds when (·) is a Lipschitz function, subject to a related restriction on h. The proof involves the construction of a Lyapunov function based on a convex combination of G-norms.

Rational approximation of the unitary exponential

Huhtanen, M. — 2009-02-16

The exponential function maps the Lie algebra of skew-Hermitian matrices to the respective Lie group of unitary matrices. Alternatively, unitary matrices are obtained by applying the generalized Cayley transform to polynomials in skew-Hermitian matrices. For a given large skew-Hermitian matrix, the related problem of choosing the polynomial to approximate its exponential applied to a vector is studied. Optimal polynomials are found with respect to various criteria.

Uniform convergence for a finite-element discretization of a viscous diffusion equation

Pierre, M. — 2009-02-09

We study a space semidiscretization of a viscous diffusion equation, obtained as a singular limit of the viscous Cahn–Hilliard equation by letting the interfacial energy tend to 0. The semidiscrete solution is shown to converge uniformly in time and space to the continuous solution, on finite time intervals, as the discretization parameter h tends to 0 (in space dimension one, two and three). We obtain an optimal error bound in space dimension one assuming only a piecewise Lipschitz regularity on the initial value. This approach allows us to obtain some counterexamples concerning lower and upper bounds of solutions to Cahn–Hilliard equations. Numerical simulations confirm the theoretical results.

Lebesgue constants for Leja points

Taylor, R., Totik, V. — 2008-12-26

It is shown that for many compact sets in the plane, the Lebesgue constant of interpolation in the first n Leja points cannot grow exponentially in n. This gives a theoretical foundation for the use of Leja points in interpolation.

The spectral problem for a class of highly oscillatory Fredholm integral operators

Brunner, H., Iserles, A., Norsett, S. P. — 2008-12-10

Let be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K₀(x, y)e^i|x–y|. We study the spectral problem for large , showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;) to the associated Fredholm integral equation f = µ f+a as -> , thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

Multistep cosine methods for second-order partial differential systems

Cano, B., Moreta, M. J. — 2008-12-10

In this paper we construct and analyse some multistep methods that integrate exactly the stiff part of a second-order partial differential equation. Much emphasis is given to symmetric methods of this type in order to deal with Hamiltonian problems. In such a way, we obtain very efficient methods because they are explicit, stable and, in the case of symmetric methods, conserve properties of the system that they imitate, as is shown in a forthcoming paper. The Gautschi method is the simplest of these methods. We analyse it here using different techniques and assumptions from those in the literature, which also allow the study of methods of higher order. In particular, a symmetric fourth-order multistep method of this type is thoroughly constructed and analysed, including its resonances and possible ways to filter them.

Computation of integral manifolds for Caratheodory differential equations

Potzsche, C., Rasmussen, M. — 2008-12-02

We derive two numerical approximation schemes for local invariant manifolds of nonautonomous ordinary differential equations (ODEs) that can be measurable in time and Lipschitzian in the spatial variable. Our approach is inspired by the previous work of Jolly & Rosa (2005 IMA J. Numer. Anal., 25, 698–725) on autonomous ODEs and based on the truncated Lyapunov–Perron operators. Both of our methods are applicable to the full hierarchy of strongly stable, stable, centre stable and the corresponding unstable manifolds, and exponential refinement strategies yield exponential convergence. Several examples illustrate our approach.

A new finite-element discretization of the Stokes problem coupled with the Darcy equations

Bernardi, C., Hecht, F., Nouri, F. Z. — 2008-12-02

The flow in a rigid porous medium with a crack is usually modelled by the Darcy equations coupled with the Stokes problem. We first propose a new variational formulation of the Stokes system, where the unknowns are the vorticity, the velocity and the pressure, and describe the corresponding finite-element discretization. We extend this discretization to the case where the Darcy and the Stokes equations are coupled and prove optimal a priori and a posteriori error estimates. We conclude with some numerical experiments.

The hp-version of the boundary element method with quasi-uniform meshes for weakly singular operators on surfaces

Bespalov, A., Heuer, N. — 2008-11-24

We prove an a priori error estimate for the hp-version of the boundary element method with weakly singular operators in three dimensions. The underlying meshes are quasi-uniform. Our model problem is that of the Laplacian exterior to an open surface, where the solution has strong singularities that are not L₂-regular. Our results confirm previously conjectured convergence rates in h (the mesh size) and p (the polynomial degree) and these rates are given explicitly in terms of the exponents of the singular functions. In particular, for sufficiently smooth given data we prove a convergence in the energy norm like O(h^1/2p^–1).

Some numerical methods for second-kind Fredholm integral equations on the real semiaxis

Mastroianni, G., Milovanovic, G. V. — 2008-11-20

In this paper we introduce some numerical methods for solving Fredholm integral equations of the second kind on the real semiaxis and prove that the proposed procedures are stable and convergent. Error estimates and numerical tests are also included.

Gauss-Hermite wave packet dynamics: convergence of the spectral and pseudo-spectral approximation

Faou, E., Gradinaru, V. — 2008-11-20

The time-dependent linear Schrödinger equation for nuclei on the whole space is semidiscretized using Hermite and Gauss–Hermite basis functions. These are well suited, on the one hand, for the conservation properties of the numerical solution and, on the other hand, for their remarkable approximation properties. We investigate theoretically and numerically the convergence of the spectral and pseudo-spectral Gauss–Hermite semidiscretization schemes.

Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Wu, H., Zhang, Z. — 2008-10-29

We utilize the recovered gradient by the polynomial-preserving recovery to enhance the eigenvalue approximation of the Laplace operator under adaptive meshes. Superconvergence rate is established and numerical tests on benchmark problems support our theoretical findings.

A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions

Liu, F., Madden, N., Stynes, M., Zhou, A. — 2008-10-03

The linear reaction–diffusion problem – ²u + bu = f is considered on the unit square with homogeneous Dirichlet boundary conditions. Here is a small positive parameter and the problem is in general singularly perturbed. The numerical solution of this problem is analysed on a Shishkin mesh that has N intervals in each coordinate direction, using the Galerkin finite-element method with bilinear trial functions. The accuracy of this method, measured in the associated energy norm, is shown to be O(N^–2 + ^1/2N^–1 ln N). It is proved that a two-scale sparse grid method achieves the same order of accuracy while reducing the number of degrees of freedom from O(N²) to O(N^3/2). These results are then generalized to systems of reaction–diffusion equations.

Convolution of hp-functions on locally refined grids

Hackbusch, W. — 2008-10-03

Usually, the fast evaluation of a convolution integral requires that the functions f and g have a simple structure based on an equidistant grid in order to apply the fast Fourier transform. Here, we discuss the efficient performance of the convolution of hp-functions in certain locally refined grids. More precisely, the convolution result is projected into some given hp-space (Galerkin approximation). The overall cost is O(p²N log N), where N is the sum of the dimensions of the subspaces containing f, g and the resulting function, while p is the maximal polynomial degree.

Numerical approximation of corotational dumbbell models for dilute polymers

Barrett, J. W., Suli, E. — 2008-09-18

We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function satisfying a Fokker–Planck-type parabolic equation. We focus on finitely extensible nonlinear elastic-type dumbbell models. In the case of a corotational drag term, we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier–Stokes–Fokker–Planck system.

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.

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.