IMA Journal of Numerical Analysis - current issue

Compact embeddings of broken Sobolev spaces and applications

Buffa, A., Ortner, C. — Tue, 27 Oct 2009 03:48:04 PDT

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.

Discontinuous Galerkin approximations for Volterra integral equations of the first kind

Brunner, H., Davies, P. J., Duncan, D. B. — Tue, 27 Oct 2009 03:48:04 PDT

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.

From high oscillation to rapid approximation III: multivariate expansions

Iserles, A., Norsett, S. P. — Tue, 27 Oct 2009 03:48:04 PDT

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.

Convergence rates for adaptive finite elements

Gaspoz, F. D., Morin, P. — Tue, 27 Oct 2009 03:48:04 PDT

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.

Numerical approximation of corotational dumbbell models for dilute polymers

Barrett, J. W., Suli, E. — Tue, 27 Oct 2009 03:48:04 PDT

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.

Convolution of hp-functions on locally refined grids

Hackbusch, W. — Tue, 27 Oct 2009 03:48:04 PDT

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.

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

Liu, F., Madden, N., Stynes, M., Zhou, A. — Tue, 27 Oct 2009 03:48:04 PDT

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.

Enhancing eigenvalue approximation by gradient recovery on adaptive meshes

Wu, H., Zhang, Z. — Tue, 27 Oct 2009 03:48:04 PDT

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.

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

Faou, E., Gradinaru, V. — Tue, 27 Oct 2009 03:48:05 PDT

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.

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

Mastroianni, G., Milovanovic, G. V. — Tue, 27 Oct 2009 03:48:05 PDT

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.