Parallel Grid Refinement and a Posteriori Error Analysis

   Stanimire Z. Tomov

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:

  1. to further develop a parallel grid generation tool for three dimensional problems;
  2. to help other researchers in CASC to use it for algorithm testing purposes;
  3. to integrate it with HYPRE data structures;
  4. to use it with HYPRE preconditioners through the Finite Element Interface (FEI);
  5. to work on a posteriori error analysis from theoretical and practical point of view;
  6. 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.