Error Control and Parallel Adaptive Grid Refinement for Convection-Diffusion-Reaction Problems

   Stanimire Z. Tomov

Texas A&M University

Motivation and Objective
Simulation of flow and transport in porous media gives rise to extremely large scale computations, making the consideration of iterative solvers natural. The solutions of such problems exhibit local behavior. Therefore it is essential that local grid refinement, based on a posteriori error analysis, is applied. Also, fundamental physical limitations on the computer processing speed may require the exploitation of parallelism. The goal is to create a tool that is based on discretization techniques utilizing finite elements/volumes, efficient preconditioning (in parallel) of the resulting sparse system, error control and adaptive grid refinement (in parallel).

Approach and Accomplishments
Our computational approach is described as follows. We use mesh generator (triangle for 2-D meshes and NETGEN for 3-D meshes) to generate a good coarse mesh. Then the considered problem is solved sequentially on the coarse mesh (by every processor). The solution is used to compute a posteriori error estimates, which are used as weights in an element based splitting of the coarse mesh into sub-domains (using METIS). Such splitting insures that the local refinements that follow will produce computational mesh with number of triangles/tetrahedrons balanced over the sub-domains. Every sub-domain is ``mapped'' to a processor. Then, based on a posteriori error analysis, each processor refines consecutively its region independently. After every step of independent refinement there is communication between the processors in order to make the mesh on that level globally conforming.
Concerning the a posteriori error analysis I worked with Dr. Raytcho Lazarov on article ``Error Control and Adaptive Grid Refinement for Convection-Diffusion-Reaction Problems in 3-D''. The article contains the description of an adaptive numerical technique based on finite volume approximations and the computational results of various model simulations of steady-state single phase flow and transport of passive chemicals in non-homogeneous porous media in 3-D.
I developed a 2-D code with functionality as described in the computational approach above. The obtained multi-level structure is used to define multigrid preconditioners. I worked with Dr. Panayot Vassilevski and Dr. Charles Tong on connecting the developed software to the hypre Preconditioner Library. The connection is implemented using The Finite-Element Interface (FEI) specification, which provides a layered abstraction that minimizes the concern for the internal details in the hypre library. The initialization is done in parallel. GUI, using Motif, has been developed to utilize the selection of different hypre options. After the solution is obtained on certain level it's send directly through AF_INET socket to visualizer (SG, developed by Dr. Michael Holst) residing on the users machine. The benefit in such strategy is that the parallel machine (usually remote) is used only for computations and the local machine handles the visualization. This idea is very efficient for real time visualization of time dependent problems.
The 3-D version of the code has the same features. Under construction is the parallel local refinement step of communication between the processors for making the mesh on certain level globally conforming. This step is significantly more complicated than its 2-D equivalent.
Raviart-Thomas zero order (RT0) finite element has been added to both the 2-D and 3-D codes. Under development is discontinuous approximation of convection terms for mixed finite element.
Both the 2-D and 3-D codes are written in C++. The parallel computations are done using the Message Passing Interface library (MPI).
I finished with Dr. Raytcho Lazarov and Dr. Panayot Vassilevski article on ``Interior penalty discontinuous approximations of elliptic problems'', which has been submitted to SIAM J. Scientific Computing.

Future Plans
I will continue my work on error control, adaptive grid refinement and a posteriori error estimates for elliptic problems based on finite volume/element methods, paying special attention to parallel computations. The 3-D parallel version of the developed code is still an on-going project.

August 17, 2000.