Dissertation Defense: Hongzhi Wan

The thesis is titled "C^0 Interior Penalty Methods for the Stream Function Formulation of the Surface Stokes Problem". This thesis develops and analyzes finite element methods for fourth-order partial differential equations posed on smooth surfaces, with particular emphasis on the stream function formulation of the surface Stokes problem. The stream function formulation replaces the velocity-pressure saddle-point system by a scalar fourth-order surface equation. This formulation automatically incorporates tangentiality and incompressibility of the velocity field, but it also introduces new challenges related to fourth-order surface differential operators, geometric approximation, and curvature-dependent terms. A central theme of this thesis is the construction of curvature-independent discretizations. To this end, we introduce a Hessian-like surface differential operator and derive integration-by-parts identities that allow the stream function formulation to be written without explicitly approximating the Gauss curvature in the principal bilinear form. This provides the analytical foundation for primal C^0 interior penalty methods based on standard continuous finite element spaces. The first method developed in this thesis is a fitted surface finite element method for the stream function formulation of the surface Stokes problem. The method is posed on polynomially approximated surfaces, uses continuous piecewise polynomial spaces for the scalar stream function, and recovers the velocity through a discrete surface curl. We prove coercivity, continuity, geometric consistency, and a priori error estimates in energy and L^2 norms. The thesis then turns to unfitted methods. We first develop a trace finite element method for the surface biharmonic equation, which serves as a model fourth-order surface problem. The method uses traces of bulk finite element functions on a discrete surface cutting through a tetrahedral background mesh, together with stabilization terms t