IMA Journal of Numerical Analysis - Advance Access

Gaussian quadrature rules using function derivatives

Milovanovic, G. V., Cvetkovic, A. S. — Fri, 20 Nov 2009 05:24:18 PST

For finite positive Borel measures supported on the real line we consider a new type of quadrature rule with maximal algebraic degree of exactness that involves function derivatives. We prove the existence of such quadrature rules and describe their basic properties. We also give an application of these quadrature rules to the solution of a Cauchy problem without solving it directly. Numerical examples are included as well.

Kantorovich's theorems for Newton's method for mappings and optimization problems on Lie groups

Wang, J.-H., Li, C. — Mon, 16 Nov 2009 06:49:15 PST

With the classical assumptions on f, a convergence criterion of Newton's method (independent of affine connections) to find zeros of a mapping f from a Lie group to its Lie algebra is established, and estimates of the convergence domains of Newton's method are obtained, which improve the corresponding results in Owren & Welfert (2000, BIT Numer. Math., 40, 121–145) and Wang & Li (2006, J. Zhejiang Univ. Sci. A, 8, 978–986). Applications to optimization are provided and the results due to Mahony (1996, Linear Algebra Appl., 248, 67–89) are extended and improved accordingly.

Linearly implicit methods for a semilinear parabolic system arising in two-phase flows

Akrivis, G., Papageorgiou, D. T., Smyrlis, Y.-S. — Mon, 09 Nov 2009 04:08:18 PST

We study the discretization of a nonlinear parabolic system arising in two-phase flows, which in a special case reduces to the Kuramoto–Sivashinsky equation, by linearly implicit methods and, in particular, by implicit–explicit multistep methods. We carry out extensive numerical experiments to investigate the accuracy and efficiency of these algorithms with extremely satisfactory results. These numerical experiments establish the analyticity of the solution and the existence of global attractors (rigorous proofs of such results for this system are not available). Our numerical experiments yield a sharp estimate for the band of analyticity of the solutions as the parameters vary. The accuracy of the schemes enables, in general, the exhaustive numerical study of such systems and the full classification of the inertial manifold. We provide numerical examples of travelling time-periodic attractors as well as quasi-periodic and chaotic attractors.

Analysis of stability and convergence of finite-difference methods for a reaction-diffusion problem on a one-dimensional growing domain

Mackenzie, J. A., Madzvamuse, A. — Mon, 09 Nov 2009 04:08:17 PST

In this paper we consider the stability and convergence of finite-difference discretizations of a reaction–diffusion equation on a one-dimensional domain that is growing in time. We consider discretizations of conservative and nonconservative formulations of the governing equation and highlight the different stability characteristics of each. Although nonconservative formulations are the most popular to date, we find that discretizations of the conservative formulation inherit greater stability properties. Furthermore, we present a novel adaptive time integration scheme based on the well-known method and describe how the parameter should be chosen to ensure unconditional stability, independently of the rate of domain growth. This work is a preliminary step towards an analysis of numerical schemes for the solution of reaction–diffusion systems on growing domains. Such problems arise in many practical areas including biological pattern formation and tumour growth.

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

Charbonneau, B., Svyrydov, Y., Tupper, P. F. — Fri, 06 Nov 2009 07:09:55 PST

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.

Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes

Ainsworth, M., Rankin, R. — Tue, 27 Oct 2009 09:39:27 PDT

We obtain fully computable constant free a posteriori error bounds on the broken energy seminorm and the discontinuous Galerkin (DG) norm of the error for nonuniform polynomial order symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin and incomplete interior penalty Galerkin finite-element approximations of a linear second-order elliptic problem on meshes containing hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and the DG norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and the DG norm of the error up to a constant and higher-order data oscillation terms.

On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion

Gerisch, A. — Fri, 23 Oct 2009 01:09:43 PDT

Recently, a nonlocal term has been introduced in time-dependent partial differential equation (PDE) models of cell migration in tissue. This term is used to model adhesive effects between cells and also between cells and the extracellular matrix. We assume periodic boundary conditions for the model and that the PDE system is discretized following the method of lines and using a finite-volume scheme on a uniform grid in space. For high-resolution simulations of the PDE system an efficient evaluation of the approximation of the nonlocal term is crucial. For one and two spatial dimensions we develop suitable approximations of the nonlocal term and evaluate these using fast Fourier transform (FFT) techniques. Comprehensive numerical tests show the accuracy and efficiency of our approach. We also demonstrate the impact of the proposed scheme for the treatment of the nonlocal term on simulation times for a differential cell adhesion model. We discuss extensions and applicability of our work to systems with nonperiodic boundary conditions and for other nonlocal PDE models from mathematical biology.

Pointwise error estimate and asymptotic error expansion inequalities for a stabilized Galerkin method

Ku, J. — Tue, 06 Oct 2009 04:07:35 PDT

This paper contains new pointwise error estimates for a stabilized Galerkin method proposed by Bramble et al. (1998, Comput. Methods Appl. Mech. Eng., 152, 195–210) and Ku (2007, Math. Comput., 76, 97–114) for second-order elliptic partial differential equations. The estimates show a local dependence of the error at a point on the derivative of the solution u and a weak dependence on the global norm. The results in this paper are stronger than the maximum norm error estimates in Ku (2007). As elementary consequences of the new pointwise error estimates, we provide asymptotic error expansion inequalities by following the idea of Schatz (1998, Math. Comput., 67, 877–899). The results are valid for large classes of finite-element spaces on irregular grids.

Componentwise error bounds for linear complementarity problems

Wang, Z., Yuan, Y.-x. — Sat, 26 Sep 2009 03:42:54 PDT

Componentwise error bounds for linear complementarity problems are presented. For the problem with an H-matrix the error bound can be computed by solving a system of linear equations. It is proved that our error bound is more accurate than that obtained recently by Chen & Xiang (2006, Math. Prog., Ser. A, 106, 513–525). Numerical results show that the new bound is often much better than previous ones.

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

Georgoulis, E. H., Houston, P., Virtanen, J. — Sat, 26 Sep 2009 03:42:54 PDT

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.

On the convergence of a wide range of trust region methods for unconstrained optimization

Powell, M.J.D. — Fri, 25 Sep 2009 05:46:31 PDT

We consider trust region methods for seeking the unconstrained minimum of an objective function F( ), , when the gradient F( ), , is available. The methods are iterative with ₁ being given. The new vector of variables _k+1 is derived from a quadratic approximation to F that interpolates F( _k) and F( _k), where k is the iteration number. The second derivative matrix of the quadratic approximation, B_k say, can be indefinite, because the approximation is employed only if the vector of variables satisfies || – _k|| ≤ _k, where _k is a "trust region radius" that is adjusted automatically. Thus the approximation is useful if || F( _k)|| is sufficiently large and if ||B_k|| and _k are sufficiently small. It is proved under mild assumptions that the condition || F( _k+1)|| ≤ is achieved after a finite number of iterations, where is any given positive constant, and then it is usual to end the calculation. The assumptions include a Lipschitz condition on F and also F has to be bounded below. The termination property is established in a single theorem that applies to a wide range of trust region methods that force the sequence F( _k), k = 1, 2, 3, ..., to decrease monotonically. Any choice of each symmetric matrix B_k is allowed, provided that ||B_k|| is bounded above by a constant multiple of k.

Numerical solution of a parabolic transmission problem

Jovanovic, B. S., Vulkov, L. G. — Thu, 10 Sep 2009 05:52:10 PDT

In this paper we investigate an initial boundary-value problem for a one-dimensional parabolic equation in two disconnected intervals. A finite-difference scheme approximating this problem is proposed and analysed. An estimate of the convergence rate, compatible with the smoothness of the input data (up to a logarithmic factor of the mesh size), is obtained.

Optimal stability for trapezoidal-backward difference split-steps

Dharmaraja, S., Wang, Y., Strang, G. — Mon, 07 Sep 2009 02:01:03 PDT

The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (t trapezoidal, (1 – )t for BDF2) retains second-order accuracy. The ‘magic choice’ allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that also gives the largest region of linearized stability.

Discretizing the Maxwell-Klein-Gordon equation by the lattice gauge theory formalism

Christiansen, S. H., Halvorsen, T. G. — Mon, 07 Sep 2009 02:01:03 PDT

In this article we study the discretization of the Maxwell–Klein–Gordon equation from a variational point of view. We first solve the problem with an action corresponding to the Yee scheme for the Maxwell part, which is automatically gauge invariant, and a gauge invariant action for the Klein–Gordon part given by the lattice gauge theory. The action is showed to be consistent with the continuous formulation, and the equations to be solved are derived from a discrete stationary action principle. Due to the gauge invariance, the local electric charge can be shown to be conserved. As this is an essential feature of the continuous model, this conservation can be viewed as the main achievement of this scheme. Thereafter we compare the above-described scheme with a scheme that uses a standard finite-difference approximation of the derivatives and where the coupling between the scalar field and the gauge field is done in the simplest way. This scheme will possess a global gauge symmetry that ensures the conservation of global charge as in the hybrid case, but the scheme has no local symmetry and no locally conserved charge. Finally, we present some numerical results in the temporal gauge, shedding light on the theoretical discussion.

A discretization of the phase mass balance in fractional step algorithms for the drift-flux model

Gastaldo, L., Herbin, R., Latche, J.-C. — Mon, 07 Sep 2009 02:01:02 PDT

We address in this paper a parabolic equation used to model the phase mass balance in two-phase flows, which differs from the mass balance for chemical species in compressible multicomponent flows by the addition of a nonlinear term of the form ·(y)u_r, where y is the unknown mass fraction, stands for the density, is a regular function such that (0) = (1) = 0 and u_r is a (not necessarily divergence free) velocity field. We propose a finite-volume scheme for the numerical approximation of this equation, with a discretization of the nonlinear term based on monotone flux functions. Under the classical assumption that the discretization of the convection operator must be such that it vanishes for a constant y, we prove the existence and uniqueness of the solution, together with the fact that it remains within its physical bounds, i.e., within the interval [0, 1]. Then this scheme is combined with a pressure correction method to obtain a semi-implicit fractional step scheme for the so-called drift-flux model. To satisfy the above-mentioned assumption a specific time stepping algorithm with particular approximations for the density terms is developed. Numerical tests are performed to assess the convergence and stability properties of this scheme.

On the stability of finite-element discretizations of convection-diffusion-reaction equations

Knobloch, P., Tobiska, L. — Thu, 27 Aug 2009 07:31:58 PDT

A priori error estimates for the local projection (LP) stabilization applied to convection–diffusion–reaction equations are generally based on the coercivity of the underlying bilinear form with respect to the LP norm. We show that the bilinear form of the LP stabilization satisfies an inf–sup condition in a stronger norm that is equivalent to that of the streamline upwind/Petrov–Galerkin method. As a consequence, we get some insight into the stabilization mechanism of Galerkin discretizations of higher order.

Quadratic projection methods for approximating the spectrum of self-adjoint operators

Strauss, M. — Thu, 27 Aug 2009 00:57:40 PDT

The pollution-free approximation of the spectrum for self-adjoint operators using a quadratic projection method has recently been studied. Higher-order pollution-free approximation can be achieved by combining this technique with a method due to Kato. To illustrate, an example from magnetohydrodynamics is considered. Whether or not this procedure converges to the whole spectrum is unknown. Combining the quadratic method with the Galerkin method, we derive procedures that do converge to the whole spectrum and without pollution.

Finite-volume schemes for noncoercive elliptic problems with Neumann boundary conditions

Chainais-Hillairet, C., Droniou, J. — Tue, 25 Aug 2009 02:22:57 PDT

We consider a convective–diffusive elliptic problem with Neumann boundary conditions. The presence of the convective term results in noncoercivity of the continuous equation and, because of the boundary conditions, the equation has a nontrivial kernel. We discretize this equation with finite-volume techniques and in a general framework that allows us to consider several treatments of the convective term, namely, via a centred scheme, an upwind scheme (widely used in fluid mechanics problems) or a Scharfetter–Gummel scheme (common to semiconductor literature). We prove that these schemes satisfy the same properties as the continuous problem (one-dimensional kernel spanned by a positive function, for instance) and that their kernel and solution converge to the kernel and solution of the partial differential equation. We also present several numerical implementations, studying the effects of the choice of one scheme or the other in the approximation of the solution or the kernel.

Householder triangularization of a quasimatrix

Trefethen, L. N. — Fri, 21 Aug 2009 04:55:08 PDT

A standard algorithm for computing the QR factorization of a matrix A is Householder triangularization. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a ‘matrix’ whose ‘columns’ are functions defined on an interval [a, b]. Applications are mentioned to quasimatrix least squares fitting, singular value decomposition and determination of ranks, norms and condition numbers, and numerical illustrations are presented using the chebfun system.

Neumann-Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum

Marletta, M. — Fri, 21 Aug 2009 04:55:07 PDT

We prove that one of the most commonly used techniques for approximating the spectra of certain classes of non-self-adjoint elliptic partial differential equations on exterior domains does not suffer from spectral pollution except possibly in the spectral gaps. This generalizes a well-known result from the self-adjoint case. We also show how the method can be used in conjunction with some simple tricks to avoid spectral pollution for the self-adjoint case. Our proofs are based on a new approach to the nesting set analysis for Neumann to Dirichlet maps first proposed by Amrein and Pearson in 2004, with enhanced convergence results obtained from an elliptic regularity bootstrapping procedure. The numerical results in Section 6 illustrate a technique for finding eigenvalues in spectral gaps of self-adjoint operators by means of a compactly supported complex shift. This method seems to be of independent interest and can be understood without reading the rest of the paper.

Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

Bultheel, A., Gonzalez-Vera, P., Hendriksen, E., Njastad, O. — Fri, 21 Aug 2009 04:55:07 PDT

This paper is concerned with rational Szego quadrature formulas to approximate integrals of the form I_µ (f) = _– f (eⁱ)dµ () by a formula such as I_n (f) = _{k = 1}ⁿ_kf (z_k), where the weights _k are positive and the nodes z_k are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szego–Radau and rational Szego–Lobatto quadrature formulas will be characterized.

A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem

Chadha, N. M., Kopteva, N. — Tue, 11 Aug 2009 08:07:15 PDT

The numerical solution of a singularly perturbed semilinear reaction–diffusion two-point boundary-value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001, SIAM J. Numer. Anal., 39, 1446–1467) to a new equation and a more intricate monitor function. It is proved that there exists a solution to the fully discrete equidistribution problem, i.e. a mesh exists that equidistributes the discrete monitor function computed from the discrete solution on this mesh. Furthermore, in the case when the boundary-value problem is linear, it is shown that after O(| ln |/ ln N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves second-order accuracy in the maximum norm, uniformly in the diffusion coefficient ². Numerical experiments are presented that support our theoretical results.

The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

Larsson, S., Saedpanah, F. — Thu, 06 Aug 2009 07:05:35 PDT

We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneous mixed Dirichlet and Neumann boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then used to prove a priori error estimates. The theory is illustrated by a numerical example.

Piecewise-smooth chebfuns

Pachon, R., Platte, R. B., Trefethen, L. N. — Tue, 28 Jul 2009 05:09:08 PDT

Algorithms are described that make it possible to manipulate piecewise-smooth functions on real intervals numerically with close to machine precision. Break points are introduced in some such calculations at points determined by numerical root finding and in others by recursive subdivision or automatic edge detection. Functions are represented on each smooth subinterval by Chebyshev series or interpolants. The algorithms are implemented in object-oriented Matlab in an extension of the chebfun system, which was previously limited to smooth functions on [–1, 1].

On the stability of fully adaptive multiscale schemes for conservation laws using approximate flux and source reconstruction strategies

Hovhannisyan, N., Muller, S. — Fri, 24 Jul 2009 05:39:39 PDT

In order to accelerate finite-volume schemes (FVSs) applied to (inhomogeneous) hyperbolic conservation laws multiresolution-based adaptive concepts can be used. The basic idea is to analyse the local regularity by means of a multiresolution analysis of cell averages. By difference information between successive refinement levels, local grid adaptation is triggered employing threshold techniques. This leads to a significant gain in computational complexity. The crucial point is to compute numerical fluxes and sources on local resolution levels such that the perturbation error introduced by the adaptive procedure due to thresholding is of the same order as the discretization error of the reference FVS on the finest uniform discretization. In the present work a modified approach based on polynomial reconstruction techniques is introduced and investigated analytically. The efficiency and accuracy of the adaptive concept is significantly improved in comparison to previous approaches, in particular, for inhomogeneous equations. This is confirmed by numerical parameter studies.

Newton-Cotes rules for Hadamard finite-part integrals on an interval

Li, B., Sun, W. — Tue, 21 Jul 2009 05:13:20 PDT

The general (composite) Newton–Cotes rules are studied for Hadamard finite-part integrals. We prove that the error of the kth-order Newton–Cotes rule is for odd k and for even k when the singular point coincides with an element junction point. Two modified Newton–Cotes rules are proposed to remove the factor ln h from the error bound. The convergence rate (accuracy) of even-order Newton–Cotes rules at element junction points is the same as the superconvergence rate at certain Gaussian points as presented in Wu & Lü (2005, IMA J. Numer. Anal., 25, 253–263) and Wu & Sun (2008, Numer. Math., 109, 143–165). Based on the analysis, a class of collocation-type methods are proposed for solving integral equations with Hadamard finite-part kernels. The accuracy of the collocation method is the same as the accuracy of the proposed even-order Newton–Cotes rules. Several numerical examples are provided to illustrate the theoretical analysis.

Numerical approximation of gradient flows for closed curves in Rd

Barrett, J. W., Garcke, H., Nurnberg, R. — Thu, 16 Jul 2009 03:36:35 PDT

We present parametric finite-element approximations of curvature flows for curves in R^d, where d ≥ 2, as well as for curves on two-dimensional manifolds in R³. Here we consider the curve shortening flow, the curve diffusion and the elastic flow. It is demonstrated that the curve shortening and the elastic flows on manifolds can be used to compute nontrivial geodesics and that the corresponding geodesic curve diffusion flow leads to solutions of partitioning problems on two-dimensional manifolds in R³. In addition, we extend these schemes to anisotropic surface energy densities. The presented schemes have very good properties with respect to stability and the distribution of mesh points, and hence no remeshing is needed in practice.

A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems

Egger, H., Schoberl, J. — Fri, 10 Jul 2009 07:17:58 PDT

We propose and analyse a new finite-element method for convection–diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin (DG) method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection–diffusion problems. By construction, the discrete solutions obtained for the limiting subproblems coincide with the ones obtained by the mixed method for the elliptic and the DG method for the limiting hyperbolic problem. We present a new type of analysis that explicitly takes into account the Lagrange multipliers introduced by hybridization. The use of adequate energy norms allows us to treat the purely diffusive, the convection-dominated and the hyperbolic regimes in a unified manner. In numerical tests we illustrate the efficiency of our approach and make a comparison with results obtained using other methods for convection–diffusion problems.

A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon

Smitheman, S. A., Spence, E. A., Fokas, A. S. — Fri, 10 Jul 2009 07:17:58 PDT

Integral representations for the solutions of the Laplace and modified Helmholtz equations can be obtained using Green's theorem. However, these representations involve both the solution and its normal derivative on the boundary, and for a well-posed boundary-value problem (BVP) one of these functions is unknown. Determining the Neumann data from the Dirichlet data is known as constructing the Dirichlet-to-Neumann map. A new transform method was introduced in Fokas (1997, Proc. R. Soc. Lond. A, 53, 1411–1443) for solving BVPs for linear and integrable nonlinear partial differential equations (PDEs). For linear PDEs this method can be considered as the analogue of the Green's function approach in the Fourier plane. In this method the Dirichlet-to-Neumann map is characterized by a certain equation, the so-called global relation, which is formulated in the complex k-plane, where k denotes the complex extension of the spectral (Fourier) variable. Here we solve the global relation numerically for the Laplace and modified Helmholtz equations in a convex polygon. This is achieved by evaluating the global relation at a properly chosen set of points in the spectral (Fourier) plane, which is why this method has been called a ‘spectral collocation method’. Numerical experiments suggest that the method inherits the order of convergence of the basis used to expand the unknown functions, namely, exponential for a polynomial basis such as Chebyshev, and algebraic for a Fourier basis. However, the condition number of the associated linear system is much higher for a polynomial basis than for a Fourier one.

Approximating piecewise-smooth functions

Lipman, Y., Levin, D. — Wed, 08 Jul 2009 07:21:49 PDT

We consider the possibility of using locally supported quasi-interpolation operators for the approximation of univariate nonsmooth functions. In such a case, one usually expects the rate of approximation to be lower than that of smooth functions. It is shown in this paper that prior knowledge of the type of ‘singularity’ of the function can be used to regain the full approximation power of the quasi-interpolation method. The singularity types may include jumps in the derivatives at unknown locations or even singularities of the form (x – s), with unknown s and . The new approximation strategy includes singularity detection and high-order evaluation of the singularity parameters, such as the above s and . Using the acquired singularity structure, a correction of the primary quasi-interpolation approximation is computed, yielding the final high-order approximation. The procedure is local, and the method is also applicable to a nonuniform data-point distribution. The paper includes some examples illustrating the high performance of the suggested method, supported by an analysis proving the approximation rates in some of the interesting cases.

Accurate approximations to time-dependent nonlinear convection-diffusion problems

de Frutos, J., Garcia-Archilla, B., Novo, J. — Mon, 06 Jul 2009 06:56:47 PDT

A new technique to improve the accuracy of the spatial discretization of nonlinear convection–diffusion equations is introduced. Both h- and p-versions of the finite-element method are considered. The procedure amounts to solving a discrete stationary convection–diffusion problem with data based on the computed standard Galerkin approximation at any fixed time. We prove that the convergence rate is increased. Numerical experiments confirm the increase in the convergence rate and show that the procedure we propose annihilates the oscillations of the Galerkin approximation in the convection-dominated regime.

Numerical approximations to the mass transfer problem on compact spaces

Gabriel, J. R., Gonzalez-Hernandez, J., Lopez-Martinez, R. R. — Thu, 02 Jul 2009 05:49:08 PDT

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. — Wed, 01 Jul 2009 08:00:54 PDT

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. — Tue, 30 Jun 2009 02:27:22 PDT

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. — Tue, 23 Jun 2009 07:30:24 PDT

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. — Tue, 23 Jun 2009 07:30:24 PDT

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. — Fri, 19 Jun 2009 07:32:27 PDT

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. — Wed, 17 Jun 2009 12:33:25 PDT

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. — Tue, 16 Jun 2009 09:52:24 PDT

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. — Mon, 08 Jun 2009 07:33:42 PDT

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. — Mon, 08 Jun 2009 07:33:41 PDT

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. — Tue, 02 Jun 2009 06:56:56 PDT

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. — Tue, 19 May 2009 15:01:38 PDT

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. — Tue, 19 May 2009 01:57:54 PDT

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. — Fri, 17 Apr 2009 12:06:58 PDT

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. — Tue, 07 Apr 2009 11:21:03 PDT

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. — Mon, 30 Mar 2009 08:23:02 PDT

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. — Mon, 30 Mar 2009 08:23:02 PDT

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. — Fri, 27 Mar 2009 07:52:04 PDT

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. — Fri, 27 Mar 2009 07:52:03 PDT

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. — Wed, 25 Mar 2009 07:30:47 PDT

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. — Mon, 16 Mar 2009 14:24:21 PDT

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. — Mon, 16 Mar 2009 14:24:20 PDT

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. — Fri, 13 Mar 2009 10:16:26 PDT

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. — Thu, 12 Mar 2009 08:37:02 PDT

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. — Fri, 06 Mar 2009 09:54:29 PST

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. — Wed, 04 Mar 2009 09:47:57 PST

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. — Fri, 27 Feb 2009 10:01:18 PST

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 .

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

Mustapha, K., Mustapha, H. — Thu, 26 Feb 2009 13:29:59 PST

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. — Thu, 26 Feb 2009 13:29:58 PST

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. — Fri, 20 Feb 2009 11:22:55 PST

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. — Mon, 16 Feb 2009 07:14:47 PST

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. — Mon, 09 Feb 2009 02:36:04 PST

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. — Fri, 26 Dec 2008 08:38:56 PST

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. — Wed, 10 Dec 2008 21:27:17 PST

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. — Wed, 10 Dec 2008 21:27:17 PST

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. — Tue, 02 Dec 2008 06:47:15 PST

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. — Tue, 02 Dec 2008 06:47:15 PST

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. — Mon, 24 Nov 2008 15:24:02 PST

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).