Covolume techniques for anisotropic media/application of spectral methods to a Cahn-Hilliard model of phase transition

This thesis contains two parts. The first part solves a continuum mechanics problem with the covolume method, and the second part solves a Cahn-Hilliard model of continuum phase transition with spectral methods. The outline is as follows.; Part I. The covolume method is a new approach to discretize and numerically solve flow problems. It works on general meshes. Central to the approach is the utilization of dual pairs of meshes that are orthogonally related in certain sense. The covolume method gives simple schemes and good approximations to the solution of div-curl systems and it works directly with the systems. With finite element methods a least squares formulation is usually needed in order to get a convergent scheme. In this part, we first present the discretization scheme and analysis for the div-curl system in isotropic media and then show that the covolume method works just as well for div {dollar}A{dollar}u = {dollar}rho{dollar}, curl u = {dollar}omega{dollar} in a region of {dollar}IRsp2{dollar}. This is a nontrivial extension, since it requires the introduction of tangential components. These components cause substantial changes in the analysis of the scheme. Our results extend to other first order elliptic systems of equations.; Part II. Spectral methods solve partial differential equations numerically. With the methods, the solution to an equation is approximated by a truncated series of eigenfunctions of certain differential operators. Spectral methods have been used in the numerical simulations of incompressible flows, compressible flows, computational metereology, etc. for many years. The application of the methods in this part of the thesis is to use a spectral Galerkin version of a Cahn-Hilliard model for continuum phase transitions with Dirichlet boundary conditions in a finite interval (1-d case). We have obtained spectral accuracy in the error estimate of the approximation based on {dollar}Lsp2{dollar}-norm. For comparison, linear finite elements can give only a second order accuracy.