Automatic and Adaptive Mesh Generation with Applications to Scientific Computing

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This project combines the expertise developed in the numerical analysis community with that developed in algorithm design community to solve scientific problems governed by partial differential equations. Most of the effort will incorporate the analytical techniques from the numerical analysis area with the combinatorial techniques from the algorithms area. Central to this project is the design and analysis of new and novel meshing algorithms well suited for use with numerical techniques such as the finite element or finite volume methods. Special attention will be given to meshing in three dimensions. In particular, extension of algorithms developed under prior NSF grants will be constructed. These extensions include analyzable methods for both mesh refinement and coarsening. The more general problem of adaptive meshing for problems that are changing with time, such as solutions to singular and degenerate parabolic equations, will also be studied. The use of triangles and tetrahedra that may have small angles but no angles approaching 180 degrees will also be undertaken. Fundamental to the project is the development of good methods to estimate the eigenvalues of graphs. These tools will be used to estimate convergence rates for preconditioned iterative system solvers, to estimate interpolation error for particular mesh algorithms, to compute error estimates for control volume methods, and to analyze spectral separator methods.