Texas A&M University

** Motivation and Objective **

Parallel grid generation tools play an important role
in the scientific research that requires the power
of high performance parallel computers. To enable the
development of efficient computational technologies
such tools may have to generate finer meshes only in
some regions of the computational domain. This could be
achieved by applying a posteriori error analysis.
The goal of my summer project was to work on this problem area.
More precisely:

- to further develop a parallel grid generation tool for three dimensional problems;
- to help other researchers in
*CASC*to use it for algorithm testing purposes; - to integrate it with
*HYPRE*data structures; - to use it with
*HYPRE*preconditioners through the Finite Element Interface (*FEI*); - to work on a posteriori error analysis from theoretical and practical point of view;
- to develop 3-D visualization.

** Approach and Accomplishments **

A parallel mesh generation tool, named *ParaGrid*, was
further developed.
The development was a continuation of a 2-D project that I started
last summer in CASC. *ParaGrid* is software that takes as input
a coarse tetrahedral mesh, which describes well the domain, splits it
using *METIS*, distributes the partitioning among the
available processors and generates in parallel a sequence of meshes.
It has it's internal solvers and is able to generate various
Finite Element/Volume discretizations. The maintained data
structures allow *ParaGrid* to be easily connected to (or
used to provide data to) external parallel finite element/volume
solvers based on domain decomposition. It has been successfully used
from several researchers in *CASC* for algorithm testing
purposes.

I worked with Charles Tong on data structures for parallel finite
element problems. The stress was on the generation of a parallel element
topological data structure. I finished the generation of
*HYPRE* *ParCSR* matrices giving the relations
``element_node'', ``element_face'', ``face_node'' and their
transposed. The test data was generated with *ParaGrid*.

Generation and solution routines for elasticity problems were added
to the code. *HYPRE* preconditioners and solvers can be used.
The connection is done trough *FEI 3.0*.

I completed with Dr. Raytcho Lazarov a study
on a posteriori error control strategies for finite volume element
approximations of second order elliptic differential equations.
These refinement techniques
were applied to the finite volume discretizations of various
boundary value problems for steady-state convection-diffusion-reaction
equations in 2 and 3 dimensions.
The results were summarized in an article on
``A posteriori error estimates for finite volume element approximations
of convection-diffusion-reaction equations'' which has been submitted
to Comput. Geosciences.

Concerning the visualization a tool, named *GLVis*, was developed.
*GLVis* started from a 2-D visualizer, that was developed in
a team project at Texas A&M University. Some of its features are
solution visualization in moving cutting plane, input from files and
*AF_INET* sockets, vector field and displacements visualization, etc.

** Future Plans **

I'll continue my work on error control, adaptive grid refinement and
a posteriori error analysis as topic of my PhD thesis.
Mesh derefinement software is under
development for the case of adaptive grid refinement for time
dependent problems.

*
August 17, 2001.
*