A divergence-conforming finite element method for the surface Stokes equation

Abstract

The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini H(div)-conforming finite element space. The resulting spaces are tangential to the surface, but require penalization of jumps across element interfaces in order to weakly maintain H 1 conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

Error analysis of higher order trace finite element methods for the surface Stokes equations

“…Beyond biological sciences, fluid equations on surfaces appear in the literature on modeling of foams, emulsions and liquid crystals; see, e.g., [42,43,13,6,35,26]. Despite the apparent practical and mathematical relevance, such systems have received little attention from the scientific computing community until the very recent series of publications [27,20,38,39,12,30,34,26,15,4,21,33,5,23] that evidences a strongly growing interest in the development and analysis of numerical methods for fluid equations posed on surfaces.…”

“…In [12], instead, surface Taylor-Hood elements are used and combined with a Lagrange multiplier method to satisfy the tangentiality constraint. Divergence-free DG and H(div)-conforming finite element methods for the surface Stokes problem were recently introduced in [23,4]. These methods enforce the tangentiality condition strongly.…”

“…First stability and error analyses of finite element formulations for the surface Stokes problem are presented only in very recent papers [4,30,33,5]. The authors of [4] present an analysis of the lowest-order Brezzi-Douglas-Marini H(div)-conforming finite element.…”

“…First stability and error analyses of finite element formulations for the surface Stokes problem are presented only in very recent papers [4,30,33,5]. The authors of [4] present an analysis of the lowest-order Brezzi-Douglas-Marini H(div)-conforming finite element. The surface Stokes problem is discretized using unfitted stabilized P 1 -P 1 elements in [30], and the trace FEM with P 2 -P 1 bulk elements has been considered in [33].…”

Finite Element Error Analysis of Surface Stokes Equations in Stream Function Formulation

“…These equations are also studied as an interesting mathematical problem in its own right in, e.g., [13,45,44,2,24,1,21]. Recently there has been a strong increase in research on numerical simulation methods for surface (Navier-)Stokes equations, e.g., [28,5,38,37,39,15,29,32,6,31,22]. By far most of these and other papers on numerical methods for surface (Navier-)Stokes equations these equations are treated in the primitive velocity and pressure variables.…”