S U M M A R YWe describe and present results from a finite-volume (FV) parallel computer code for forward modelling the Maxwell viscoelastic response of a 3-D, self-gravitating, elastically compressible Earth to an arbitrary surface load. We implement a conservative, control volume discretization of the governing equations using a tetrahedral grid in Cartesian geometry and a low-order, linear interpolation. The basic starting grid honours all major radial discontinuities in the Preliminary Reference Earth Model (PREM), and the models are permitted arbitrary spatial variations in viscosity and elastic parameters. These variations may be either continuous or discontinuous at a set of grid nodes forming a 3-D surface within the (regional or global) modelling domain. In the second part of the paper, we adopt the FV methodology and a spherically symmetric Earth model to generate a suite of predictions sampling a broad class of glacial isostatic adjustment (GIA) data types (3-D crustal motions, long-wavelength gravity anomalies). These calculations, based on either a simple disc load history or a global Late Pleistocene ice load reconstruction (ICE-3G), are benchmarked against predictions generated using the traditional normal-mode approach to GIA. The detailed comparison provides a guide for future analyses (e.g. what grid resolution is required to obtain a specific accuracy?) and it indicates that discrepancies in predictions of 3-D crustal velocities less than 0.1 mm yr −1 are generally obtainable for global grids with ∼3 × 10 6 nodes; however, grids of higher resolution are required to predict large-amplitude (>1 cm yr −1 ) radial velocities in zones of peak postglacial uplift (e.g. James bay) to the same level of absolute accuracy. We conclude the paper with a first application of the new formulation to a 3-D problem. Specifically, we consider the impact of mantle viscosity heterogeneity on predictions of present-day 3-D crustal motions in North America. In these tests, the lateral viscosity variation is constructed, with suitable scaling, from tomographic images of seismic S-wave heterogeneity, and it is characterized by approximately 2 orders of magnitude (peak-to-peak) lateral variations within the lower mantle and 1 order of magnitude variations in the bulk of the upper mantle (below the asthenosphere). We find that the introduction of 3-D viscosity structure has a profound impact on horizontal velocities; indeed, the magnitude of the perturbation (of order 1 mm yr −1 ) is as large as the prediction generated from the underlying (1-D) radial reference model and it far exceeds observational uncertainties currently being obtained from space-geodetic surveying. The relative impact of lateral viscosity variations on predicted radial motions is significantly smaller.

SUMMARYA new collocation method based on quadratic splines is presented for second order two-point boundary value problems. First, O(h4) approximations to the first and second derivative of a function are derived using a quadratic-spline interpolant of u. Then these approximations are used to define an O(h4) perturbation of the given boundary value problem. Second, the perturbed problem is used to define a collocation approximation at interval midpoints for which an optimal O(h3 -j ) global estimate for the jth derivative of the error is derived. Further, O(h4-j) error bounds for the jth derivative are obtained for certain superconvergence points. It should be observed that standard collocation at midpoints gives O(h2 -J ) bounds. Results from numerical experiments are reported that verify the theoretical behaviour of the method.

Summary. We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard Cannulation of these methods leads to non-optimal approximations. In order (0 derive optimal QSC approximations, high order perturbations of the PDE problem arc generated. These perturbations can be applied either to die POE problem operators or to the right sides, thus leading 10 two different fonnulations of optimal QSC methods. The convergence properties of the QSC methods arc studied. Optimal 0 (h J-j) global error estimates for the j.th partial derivative are obtained for a certain class of problems. Moreover, error bounds for the j-th partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of Lhe QSC methods. Performance results also show that the QSC methods arc very effective from the computational point of view. The QSC methods have been implemented efficiently on parallel machines.Key words. spline collocation, elliptic partial differential equations, second order boundary value problems.AMS(MOS) subject classifications. 65N35, 65N 15.Abbreviated title.Quadratic spline collocation methods for elliptic PDEs.

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We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency of the methods. Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries.