Research Report: Manuscript CMCU-98-03, August 1998
High-order accurate spectral elements for wave problems.
Lonny L. Thompson and Sunil K. Challa
Computational Mechanics at Clemson University
Department of Mechanical Engineering
Clemson, South Carolina 29634-0921
Abstract
In order to accurately resolve waves governed by the Helmholtz equation,
the standard low-order Galerkin finite elements with piecewise linear
approximation require at least ten elements per wavelength. For large
frequencies, accurately resolving the resulting small acoustic wavelengths
may lead to prohibative computational cost in both memory and solution times.
Furthermore, a global pollution error increases with increasing
wavenumber, even when the number of elements per wavelength is held constant.
In this work, spectral elements (both linear and quadratic approximations)
are developed which significantly reduce
both local approximation (dispersion) error and global pollution effects.
The elements employ optimally selected quadrature points
and associated weighting parameters for the numerical integration of the
element arrays eminating from the discretization of the Galerkin form of
the Helmholtz problem. The resulting dynamic stiffness arrays vary only
as linear combinations with respect to frequency, and therefore can be
used for the corresponding direct time-domain solution of the wave equation.
Numerical comparisons with Galerkin and Generalized Galerkin
finite elements, and finite-difference methods, show greatly improved
accuracy with corresponding reduction in pollution.
PDF File of Report CMCU-98-03