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

Abstract

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

Appendix

Below we verify that the sequence

is exact on triangulations of contractible domains. We do so by making three observations:

• 1. If $v \in V$ and $\textrm{d}v=0$, then clearly $v=0$ by the boundary conditions and the interelement continuity constraints imposed on functions in V.

• 2. The map $\textrm{d} : W \rightarrow X$ is surjective for the following reason. On each $T \in \mathcal {T}_h$, the map

  $$\begin{aligned} {{\,\textrm{div}\,}}: H^1_0(T) \otimes \mathbb {R}^2 \rightarrow L^2_{\int =0}(T) \end{aligned}$$

  is surjective [22, Lemma B.69, p. 492], where $H^1_0(T) = \{f \in H^1(T) \mid f = 0 \text { on } \partial T \}$ and $L^2_{\int =0}(T) = \{f \in L^2(T) \mid \int _T f \, \omega = 0 \}$. By rotating vectors $90^\circ$ and identifying them with one-forms, we see that

  $$\begin{aligned} \textrm{d} : H^1_0\Lambda ^1(T) \rightarrow L^2_{\int =0}\Lambda ^2(T) \end{aligned}$$

  is surjective, where $H^1_0\Lambda ^1(T)$ denotes the space of one-forms on T with coefficients in $H^1_0(T)$ and $L^2_{\int =0}\Lambda ^2(T)$ denotes the space of square-integrable two-forms on T with vanishing integral. Now, let $F \in X$ be arbitrary. We can write $F = F_0 + F_1$, where $\int _T F_0$ vanishes on each $T \in \mathcal {T}_h$ and $F_1$ is piecewise constant. The two-form $F_0$ is in the range of $\textrm{d} : W \rightarrow X$, since we can construct $\alpha _0 \in \prod _{T \in \mathcal {T}_h} H^1_0\Lambda ^1(T) \subset W$ satisfying $\textrm{d}\alpha _0=F_0$ by above. The two-form $F_1$ is also in the range of $\textrm{d} : W \rightarrow X$, since $\textrm{d}$ maps the Whitney one-forms with vanishing trace surjectively onto the piecewise constant two-forms with vanishing mean. Thus, F is in the range of $\textrm{d} : W \rightarrow X$.

• 3. Now, consider a one-form $\alpha \in W$ satisfying $\textrm{d}\alpha =0$. We will show that there exists $v \in V$ such that $\textrm{d}v=\alpha$. The canonical Whitney interpolant of $\alpha$, being closed, belongs to the range of $\textrm{d} : V \rightarrow W$; it is the image under d of a continuous, piecewise linear function (a Whitney zero-form). So it suffices to focus on the case where $\int _e \alpha = 0$ for every $e \in \mathcal {E}_h$. On each triangle $T \in \mathcal {T}_h$, $\alpha \big |_T$ is a closed one-form belonging to $H^1\Lambda ^1(T)$, so we can construct $v_T \in H^2(T)$ such that $\textrm{d}v_T = \alpha \big |_T$ [19, Theorem 1.1] and (by adding a suitable constant) $v_T$ vanishes at one of the vertices of T. Since $\int _e \textrm{d}v_T = \int _e \alpha \big |_T = 0$ along each edge e of T, $v_T$ in fact vanishes at every vertex of T. On any edge e shared by two triangles $T_1$ and $T_2$, the equality

  $$\begin{aligned} \textrm{d}i_{T_1,e}^* v_{T_1} = i_{T_1,e}^* \textrm{d}v_{T_1} = i_{T_1,e}^* \alpha = i_{T_2,e}^* \alpha = i_{T_2,e}^* \textrm{d}v_{T_2} = \textrm{d}i_{T_2,e}^* v_{T_2}, \end{aligned}$$

  together with the fact that $v_{T_1}$ and $v_{T_2}$ vanish at the endpoints of e, ensures that the trace of $v_{T_1}$ agrees with that of $v_{T_2}$ everywhere along e. By similar reasoning, v (the function whose restriction to T is $v_T$ for each $T \in \mathcal {T}_h$) vanishes on edges $e \in \mathcal {E}_h \setminus \mathcal {E}_h^0$. It follows that $v \in V$ and $\alpha =\textrm{d}v$.