ylfink@water.waterloo.edu (ylfink) (11/01/88)
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF WATERLOO
SEMINAR ACTIVITIES
SCIENTIFIC COMPUTATION SEMINAR
- Thursday, November 3, 1988
Dr. Joseph E. Flaherty, Rensselaer Polytechnic Institute,
Troy, New York, will speak on ``Adaptive Methods for Time-
Dependent Partial Differential Equations''.
TIME: 4:00 PM
ROOM: DC 1304
ABSTRACT
We discuss adaptive local refinement procedures for solving
vector systems of time-dependent partial differential
equations in one and two space dimensions. Each adaptive
algorithm contains a basic finite element or finite volume
procedure to compute approximate solutions of the
differential system on a given mesh, an error indication
procedure to decide which regions of the problem domain
require greater resolution, and an adaptive feedback
strategy to generate a finer mesh when accuracy is not
sufficient. Finite element and finite volume procedures
utilizing piecewise polynomial approximations in space and
either explicit or implicit finite difference methods in
time will be discussed.
Temporal mesh refinement strategies can either be local or
global. Global temporal refinement strategies lead to
adaptive methods of lines. Such schemes can fully utilize
the power of existing software for solving initial value
problems for ordinary differential equations. Local
temporal refinement strategies, on the other hand, offer the
promise of greater efficiency by using different time steps
on different portions of the problem domain.
Spatial refinement strategies are grouped into three
categories: h-refinement where a mesh is refined, but the
order of the finite volume or finite element method is
unchanged; p-refinement where the same mesh is used for
methods of increasing order of accuracy; and r-refinement
where the structure of a mesh is changed without either
increasing the number of cells or the order of the method.
Mesh refinement can further be classified as being cellular
or noncellular. With cellular refinement, the computational
cells of finer meshes are properly nested within the
boundaries of cells of coarser meshes; thus, simplifying the
transfer of solutions between meshes having different
spacings. With noncellular refinement, the cells of finer
meshes can overlap those of coarser ones. Mesh structures
can be kept uniform and this may offer improved performance
on computers having vector processors. In either case, a
tree structure, with finer grids regarded as offspring of
coarser ones, is used to manage the data associated with the
refinement process. Several specific cellular and
noncellular refinement techniques will be presented. The
use of these procedures in combination with r-refinement and
p-refinement techniques and with mesh generation codes will
also be discussed.
Error indicators based on estimates of the local
discretization error are used to control the adaptive
refinement process. Several error estimates for finite
element methods that utilize a hierarchic p-refinement
approach will be presented. There is an interesting
dichotomy between the errors in odd- and even-order finite
elements. In particular norms, errors in odd-order finite
element solutions are concentrated near element edges,
whereas the errors in even-order methods are principally in
element interiors. These results are used to improve the
computational efficiency of the error estimates.