No abstract

We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a fairly general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the Steepest Descent method, but it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. The approach is compared with the oscillatory integration techniques recently developed by Iserles and Nørsett.

We consider the solution of parabolic partial differential equations (PDEs). In standard time-stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step.By contrast, the method presented in this paper treats the whole of the space-time problem simultaneously. Thus the multigrid operations of smoothing and coarse-grid correction are defined on all of the space-time variables of a given grid level. The method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil. Numerical results for the one-and two-dimensional heat equations are presented and are shown to agree closely with predictions from Fourier mode analysis.

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature of high-frequency scattering problems necessitates a large number of unknowns in classical boundary element methods. In addition, the corresponding discretization matrix of the integral equation is dense. We formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies. The method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns, but retains accuracy for lower frequencies. New in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals. As a result, we obtain a sparse discretization matrix for the oscillatory problem. Moreover, numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency. The sparse discretization applies to problems where the phase of the solution can be predicted a priori, for example in the case of smooth and convex scatterers.

This paper presents an overview and comparison of iterative solvers for linear stochastic partial differential equations (PDEs). A stochastic Galerkin finite element discretization is applied to transform the PDE into a coupled set of deterministic PDEs. Specialized solvers are required to solve the very high-dimensional systems that result after a finite element discretization of the resulting set. This paper discusses one-level iterative methods, based on matrix splitting techniques; multigrid methods, which apply a coarsening in the spatial dimension; and multilevel methods, which make use of the hierarchical structure of the stochastic discretization. Also Krylov solvers with suitable preconditioning are addressed. A local Fourier analysis provides quantitative convergence properties. The efficiency and robustness of the methods are illustrated on two nontrivial numerical problems. The multigrid solver with block smoother yields the most robust convergence properties, though a cheaper point smoother performs as well in most cases. Multilevel methods based on coarsening the stochastic dimension perform in general poorly due to a large computational cost per iteration. Moderate size problems can be solved very quickly by a Krylov method with a mean-based preconditioner. For larger spatial and stochastic discretizations, however, this approach suffers from its nonoptimal convergence properties.