A tangential and penalty-free finite element method for the surface Stokes problem

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 $H({\rm div})$ 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 $\mathbb{P}^2-\mathbb{P}^1$ elements

Finite Element Methods for the Laplace-Beltrami Operator

Partial differential equations posed on surfaces arise in a number of applications. In this survey we describe three popular finite element methods for approximating solutions to the Laplace-Beltrami problem posed on an $n$-dimensional surface $\gamma$ embedded in $\mathbb{R}^{n+1}$: the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface $\Gamma$ whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface $\Gamma$ in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface $\gamma$ is embedded in an $n+1$-dimensional domain $\Omega$ which has itself been triangulated. An $n$-dimensional approximate surface $\Gamma$ is then constructed roughly speaking by interpolating $\gamma$ over the triangulation of $\Omega$, and the finite element space over $\Gamma$ consists of the trace (restriction) of a standard finite element space on $\Omega$ to $\Gamma$. In the narrow band method the PDE posed on the surface is extended to a triangulated $n+1$-dimensional band about $\gamma$ whose width is proportional to the diameter of elements in the triangulation. In all cases we provide optimal a priori error estimates for the lowest-order finite element methods, and we also present a posteriori error estimates for the parametric and trace methods. Our presentation focuses especially on the relationship between the regularity of the surface $\gamma$, which is never assumed better than of class $C^2$, the manner in which $\gamma$ is represented in theory and practice, and the properties of the resulting methods

Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters

Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai et. al. (2014), who proved convergence and optimality of AFEM for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (non-computable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. Gallistl (2015) used a similar tool in order to prove that a standard adaptive FEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or non-constant diffusion coefficients, however, the arguments of Dai et. al. and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this note we provide this missing key step, thus showing that standard adaptive FEM for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.Comment: 10 page