Analysis of trace finite element methods for surface partial differential equations

In this paper we consider two variants of a trace finite element method for solving elliptic partial differential equations on a stationary smooth manifold |Γ|⁠. A discretization error analysis for both methods in one general framework is presented. Higher-order finite elements are treated and rather general numerical approximations |Γ ₕ| of the manifold |Γ| are allowed. Optimal-order discretization error bounds are derived. Furthermore, the conditioning of the stiffness matrices is studied. It is proved that for one of these two variants the corresponding scaled stiffness matrix has a condition number |~ h⁻²|⁠, independently of how |Γ ₕ| intersects the outer triangulation.