A Trace Finite Element Method for Vector-Laplacians on Surfaces

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included

Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality

Let \Sigma be a compact surface of type (g, n), n > 0, obtained by removing n disjoint disks from a closed surface of genus g. Assuming \chi(\Sigma)<0, we show that on \Sigma, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C^\infty topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak \cite{OPS3} for type (0,n) surfaces, whose examples include bounded plane domains. Our main ingredients are as following. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on \Sigma. Secondly, we show that the space of such metrics is homeomorphic (in the C^\infty-topology) to the space of flat metrics (on \Sigma) with constantly curved boundary. Because of this, we next reduce the complicated degenerations of flat metrics to the simpler and well-known degenerations of hyperbolic metrics, and we show that determinants of Laplacians of flat metrics on \Sigma, with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because Khuri \cite{Kh} showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when \Sigma is of type (g, n), g>0; while Osgood, Phillips, and Sarnak \cite{OPS3} showed the properness when g=0.Comment: Further Revised. A technical error is corrected; the sections devoted to the proof of the insertion lemma and the separation of variables method are completely rewritten. (Sections 4, 5, and 6 in this revised version.) A lot of changes, corrections, and improvements are made throughout the paper. No mathematical change in the main theorems listed in the introductio