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We propose aC⁰interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achievedviaa novel integration-by-parts formula for the surface biharmonic operator. 

Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and $H^1$ conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor–Hood, Scott–Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not $H^1$-conforming, but do lie in $H(div)$ and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in $$L_2$$ via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor–Hood elements. 

Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $$L^{2}(H^{1})$$-type norm for the pressure component. 

We construct conforming finite element elasticity complexes on Worsey–Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey–Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.