Applications of Generic Interpolants In the Investigation and Visualization of Approximate Solutions of PDEs on Coarse Unstructured Meshes Hassan Goldani Moghaddam Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2010 In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also

Using a Difierential Equation Interpolant (DEI), one can accurately approx- imate the solution of a Partial Difierential Equation (PDE) at ofi-mesh points. The idea is to allocate a multi-variate polynomial to each mesh element and consequently, the collection of such polynomials over all mesh elements will deflne a piecewise polynomial approximation. In this paper we will investigate such interpolants on a three-dimensional unstructured mesh. As reported in (1), for a tetrahedron mesh in three dimensions, tensor product tri-quadratic and pure tri-cubic interpolants are the most appropriate candidates. We will report on the efiectiveness of these alternatives on some typical PDEs.