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Adaptive Numerical Methods

by admin last modified 2004-10-15 12:22

Adaptive Numerical Methods for Diffusion and Transport Equations in Heterogeneous Media on Distorted Polyhedral Meshes

Efficient numerical methods for the diffusion and radiation transport equations in highly heterogeneous media on general distorted polyhedralmeshes is an important topic for scientists and engineers working in computer simulation of complex physical phenomena. This statement is very relevant to several research groups at LANL, for instance, to the T-7 and CCS-4 groups, and at UH.

The project is based on the results of very successful cooperation between researchers at LANL and at UH. In 2002-2003, Yu. Kuznetsov conceived of a fundamentally new approach for solving the diffusion equations on general polygonal and polyhedral meshes by the mixed finite element method. In 2003-2004, the idea of this method was applied by researchers from LANL (M. Shashkov, J. Morel, and K. Lipnikov) and UH (Yu. Kuznetsov) to design new accurate and physically consistent mimetic discretizations based on the support operator method for the diffusion equations on polygonal meshes.  The resulting method represents a genuine breakthrough in the numerical solution of the diffusion equations on arbitrary polygonal meshes including locally refined (AMR) and nonmatching ones. The method is slated for implementation in certain ASC projects at LANL.  Extension of the method to polyhedral meshes with application to 3D diffusion equations has been done recently (FY 2004).

In this project (FY 2005), we plan to continue joint research on development and investigation of the proposed methods as well as on implementation aspects of the method and LANL relevant applications.

Long term goal:  The major long term goal of the project is to develop, investigate and evaluate on the test problems relevant to LANL applications new adaptive mimetic compatible discretizations and efficient parallel multilevel preconditioners/solver for the diffusion and radiation transport equations in heterogeneous media on strongly distorted polyhedral meshes.

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