Ph.D Thesis

Robust and Accurate Algorithms for Solving Anisotropic Singularities

This dissertation is focused on the construction of robust and accurate algorithms for mathematical models of physical phenomena that exhibit strong anisotropies, that is, when the quantities have very slow and smooth variations in some directions but have rapid variations in other directions.

Our first result is on the mathematically characterization of optimal or nearly optimal meshes for a general function which could be either isotropic or anisotropic. We give an interpolation error estimate for the continuous and piecewise linear nodal interpolation. Roughly speaking, a nearly optimal mesh is a quasi-uniform triangulation under some new metric defined by the Hessian matrix of the object function. We also prove the error estimate is optimal for strictly convex (or concave) functions.

Based on the interpolation error estimates, we introduce a new concept Optimal Delaunay Triangulation (ODT) and present practical algorithms to construct such nearly optimal meshes. By minimizing the interpolation error globally or locally, we obtain some new functionals for the moving mesh method and several new mesh smoothing schemes.

We then apply our mesh adaptation algorithms to the convection dominated convection-diffusion problems which present anisotropic singularities such as boundary layers. We develop a robust and accurate adaptive finite element method for convection dominated problems by the homotopy of the diffusion parameter.

We give an error analysis of a one dimensional convection dominated convection-diffusion problem that is discretized by the standard finite element method on layer-adapted grids. We find that it is not uniform stable with respect to the perturbation of grid points. We then design a special streamline diffusion finite element method and prove the uniform stability and optimality of our new method.

We also discuss some related concepts and problems on the optimal Delaunay triangulations.

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