Seminar Abstract
Center for Data Intensive Computing

Adaptive Methods and Their Parallel Implementation

The use of adaptive methods in the numerical solution of PDEs has two purposes: (1) to get reliable control of the error coming from the discretization of the considered PDE problem, and (2) to use the available computational resources efficiently. Adaptive methods achieve these goals by local grid refinement, based on a posteriori error analysis. Of great interest are both the theoretical and computational aspects of the methods.

Concerning the theory, I'll present the results of a study on a posteriori error control strategies for finite volume element approximations of second order elliptic differential equations. Techniques known from the finite element method are adapted to the finite volume discretizations of various boundary value problems for steady-state convection-diffusion-reaction equations. One possible application of such problems is simulation of fluid flow and transport in porous media. Locally conservative approximation schemes and methods with a posteriori error control play important role for such problems. Finite volume methods ensure such local mass conservation and combined with some upwind strategies give monotone solutions. The a posteriori error analysis takes into account the used upwinding.

Concerning the computations, I'll discuss the tools that we used, developed and implemented in a computer system for simulation of parallel adaptive methods. Some of the used tools (software) are 2/3D mesh generators (triangle, NETGEN, TrueGrid), partitioning and load balancing software (METIS), iterative solvers and preconditioners (Hypre), MPI, OpenGL, etc. The developed tools, that will be discussed, are a multilevel parallel mesh generator (ParaGrid) and a 2/3D visualizer (GLVis). ParaGrid takes as input a coarse tetrahedral mesh, splits it using Metis, generates and maintains in parallel a sequence of meshes and data structures suitable for domain decomposition and multigrid type preconditioners. GLVis features solution visualization in moving cutting plane, input from parallel machines through sockets, vector field and displacements visualization, etc.

The theory is numerically confirmed and its efficiency will be illustrated on several computational tests in 2 and 3 dimensions.