Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions

We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations oriented two-dimensional manifolds. Our construction relies Regge elements, which are piecewise polynomial symmetric (0,2)-tensor fields possessing single-valued tangential-tangential components along interfaces. When used to discretize Riemannian metric tensor, these tensor do not possess enough regularity define connections in classical sense, but we show how make sense quantities a distributional sense. then that converge certain dual Sobolev norms their smooth counterparts under refinement triangulation. also discuss projections onto spaces. relevant projection operators commute with linearized differential operators, yielding commutative diagram complexes.