Error Estimates for Finite Element/Volume Approximations of Dissipative Partial Differential Equations on Surfaces.

Numerical solutions of partial differential equations (PDEs) on arbitrary surfaces or two dimensional Riemannian manifolds are needed in diverse applications such as fluid dynamics, weather forecast and climate modeling, chemical coating, cell membrane modeling and image processing. Many discretization techniques developed for these type of problems are based on finite element methods or finite difference methods, including direct discretizations on surface meshes or discretizations via level set techniques for implicitly defined surfaces. On the other hand, finite volume methods for the numerical solution of partial differential equations have also been gaining popularity in recent decades due to their discrete conservation properties. In this work, our study is based on surface discretization using level set techniques. After presenting all the necessary background knowledge, e.g. Laplace-Beltrami operator, finite dimensional discretization and Sobolev space defined on that, we mainly focus our research on finite volume and finite element methods for elliptic equations and Cahn-Hilliard equations. For elliptic equations, we hire finite volume method to do the analysis. First, we study the fourth order elliptic equations, and we use mixed scheme to do the job. Though many theoretical investigations have focused on finite volume methods for first- and second- order partial differential equations, there is relatively little discussion on the analysis of finite volume methods applied to higher order PDEs, especially for high order PDEs defined on general surfaces. Due to the lack of comprehensive theoretical study, there have often been concerns that direct discretizations of high order PDEs based on surface triangulations may require tremendous computational effort for varying geometries and it is not clear how higher order geometric characteristics, such as the derivatives of curvatures, are to be well represented on triangulated surfaces. Our study is aimed at filling in such a gap. The second contribution of this work is that we introduced a posteriori error estimator for finite volume method of elliptic equations on surfaces. Recently, a posteriori error estimates of finite element methods for discretizing the Laplace-Beltrami operator on surfaces were rigorously analyzed, while similar studies for finite volume methods are currently lacking as far as we know. In this part, we rigorously derived a residual-based explicit a posteriori error estimator (in the sense of energy norm) for the finite volume discretization of the elliptic equations defined on a smooth surface. The last contribution of this dissertation is the error estimate of a fully discrete finite element scheme for the Cahn-Hilliard equation on surfaces.