Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations

Abstract

In this paper we study parametric TraceFEM and parametric SurfaceFEM (SFEM) discretizations of a surface Stokes problem. These methods are applied both to the Stokes problem in velocity-pressure formulation and in stream function formulation. A class of higher order methods is presented in a unified framework. Numerical efficiency aspects of the two formulations are discussed and a systematic comparison of TraceFEM and SFEM is given. A benchmark problem is introduced in which a scalar reference quantity is defined and numerically determined.

Diffusion of tangential tensor fields: numerical issues and influence of geometric properties

“…In ISFEM the orthogonal basis { t1 , t2 } is employed. The other three methods SFEM, TraceFEM, and DI use a representation of the surface gradient based on the standard gradient in R 3 .…”

“…Previous comparisons of the different methods have shown advantageous properties of SFEM with respect to accuracy and computational effort, see [3]. In order to provide numerical reference data, we use SFEM with a higher resolution in space and time and a higher polynomial order of the solution space.…”

Tangential Errors of Tensor Surface Finite Elements

“…This problem is related to the minimization of the full covariant H 1 -norm of tensor fields and has applications in the area of nematic Liquid crystals described by surface Qtensor fields, e.g., [3,32,34,35], orientation field computation on flexible manifolds and shells, e.g., [33,40,36], the time-discretisation of the vector and tensor-heat equation, e.g., [7,8,41], and surface Stokes and Navier-Stokes equations, e.g., [38,39,15,25,18,23,26,27,4,43].…”