ABSTRACT |
In this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition, and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a C2 connected (not necessarily simply connected) oriented hypersurface Γ⊂ R3 without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space R3. These representations are very convenient for the implementation of numerical methods for
surface partial differential equations. We introduce surface H(divΓ) and H(curlΓ) spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz
decomposition, in terms of these surface differential operators, based on elementary differential calculus.
As a corollary of this decomposition we obtain that for a simply connected surface, to every
tangential divergence free velocity field there corresponds a unique scalar stream function. Using this
result the variational form of the surface Stokes equation can be reformulated as a well-posed variational
formulation of a fourth order equation for the stream function. The latter can be rewritten
as two coupled second order equations, which form the basis for a finite element discretization. A
particular finite element method is explained and results of a numerical experiment with this method
are presented. |