Finite-element Schemes (finite-element + scheme)

Distribution by Scientific Domains


Selected Abstracts


A class of parallel multiple-front algorithms on subdomains

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2003
A. Bose
Abstract A class of parallel multiple-front solution algorithms is developed for solving linear systems arising from discretization of boundary value problems and evolution problems. The basic substructuring approach and frontal algorithm on each subdomain are first modified to ensure stable factorization in situations where ill-conditioning may occur due to differing material properties or the use of high degree finite elements (p methods). Next, the method is implemented on distributed-memory multiprocessor systems with the final reduced (small) Schur complement problem solved on a single processor. A novel algorithm that implements a recursive partitioning approach on the subdomain interfaces is then developed. Both algorithms are implemented and compared in a least-squares finite-element scheme for viscous incompressible flow computation using h - and p -finite element schemes. Copyright © 2003 John Wiley & Sons, Ltd. [source]


A finite-element scheme for the vertical discretization of the semi-Lagrangian version of the ECMWF forecast model

THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 599 2004
A. Untch
Abstract A vertical finite-element (FE) discretization designed for the European Centre for Medium-Range Weather Forecasts (ECMWF) model with semi-Lagrangian advection is described. Only non-local operations are evaluated in FE representation, while products of variables are evaluated in physical space. With semi-Lagrangian advection the only non-local vertical operations to be evaluated are vertical integrals. An integral operator is derived based on the Galerkin method using B-splines as basis functions with compact support. Two versions have been implemented, one using piecewise linear basis functions (hat functions) and the other using cubic B-splines. No staggering of dependent variables is employed in physical space, making the method well suited for use with semi-Lagrangian advection. The two versions of the FE scheme are compared to finite-difference (FD) schemes in both the Lorenz and the Charney,Phillips staggering of the dependent variables for the linearized model. The FE schemes give more accurate results than the two FD schemes for the phase speeds of most of the linear gravity waves. Evidence is shown that the FE schemes suffer less from the computational mode than the FD scheme with Lorenz staggering, although temperature and geopotential are held at the same set of levels in the FE scheme too. As a result, the FE schemes reduce the level of vertical noise in forecasts with the full model. They also reduce by about 50% a persistent cold bias in the lower stratosphere present with the FD scheme in Lorenz staggering (i.e. the operational scheme at ECMWF before its replacement by the cubic version of the FE scheme described here) and improve the transport in the stratosphere. Copyright © 2004 Royal Meteorological Society [source]


On accuracy of the finite-difference and finite-element schemes with respect to P -wave to S -wave speed ratio

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2010
Peter Moczo
SUMMARY Numerical modelling of seismic motion in sedimentary basins often has to account for P -wave to S -wave speed ratios as large as five and even larger, mainly in sediments below groundwater level. Therefore, we analyse seven schemes for their behaviour with a varying P -wave to S -wave speed ratio. Four finite-difference (FD) schemes include (1) displacement conventional-grid, (2) displacement-stress partly-staggered-grid, (3) displacement-stress staggered-grid and (4) velocity,stress staggered-grid schemes. Three displacement finite-element schemes differ in integration: (1) Lobatto four-point, (2) Gauss four-point and (3) Gauss one-point. To compare schemes at the most fundamental level, and identify basic aspects responsible for their behaviours with the varying speed ratio, we analyse 2-D second-order schemes assuming an elastic homogeneous isotropic medium and a uniform grid. We compare structures of the schemes and applied FD approximations. We define (full) local errors in amplitude and polarization in one time step, and normalize them for a unit time. We present results of extensive numerical calculations for wide ranges of values of the speed ratio and a spatial sampling ratio, and the entire range of directions of propagation with respect to the spatial grid. The application of some schemes to real sedimentary basins in general requires considerably finer spatial sampling than usually applied. Consistency in approximating first spatial derivatives appears to be the key factor for the behaviour of a scheme with respect to the P -wave to S -wave speed ratio. [source]


Multidimensional FEM-FCT schemes for arbitrary time stepping

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003
D. Kuzmin
Abstract The flux-corrected-transport paradigm is generalized to finite-element schemes based on arbitrary time stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non-linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions. Copyright © 2003 John Wiley & Sons, Ltd. [source]