Rigorous convergence analysis of the nodal grad-conforming construction

Establish a rigorous convergence theory for the nodal change-of-basis construction that glues grad-conforming vector-valued finite element spaces across chart interfaces on manifolds, including the approximation behavior of the resulting inter-chart continuity defects.

Background

The paper constructs grad-conforming vector-valued finite element spaces on multi-chart manifolds by modifying local nodal bases with transmission maps between incompatible coordinate frames. The resulting discrete fields are exactly tangent to the manifold and agree at interface nodes, but their traces generally exhibit jumps between nodes when the transmission map varies along an interface.

Numerical experiments for the surface Stokes problem show optimal rates in a discontinuous Galerkin-like norm, and the paper derives an interface-jump estimate. However, a rigorous convergence analysis for the nodal construction itself is not established, leaving open the proof of the observed approximation and convergence behavior.