Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature

We study numerical approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in t he principal curvatures of a two-dimensional surface. Beside the well-known Willmo re and Helfrich flows we will also consider flows involving the Gaussian curvature of the surface. Boundary conditions for these flows are highly nonlinear, and we use a v ariational approach to derive weak formulations, which naturally can be discretiz ed with the help of a mixed finite element method. Our approach uses a parametric finite e lement method, which can be shown to lead to good mesh properties. We prove st ability estimates for a semidiscrete (discrete in space, continuous in time) vers ion of the method and show existence and uniqueness results in the fully discrete case . Finally, several numerical results are presented involving convergence tests as well a s the first computations with Gaussian curvature and/or free or semi-free boundary c onditions