Distributional Gauss curvature (analysis)

After we defined the distributional Gauss curvature and its lifting and observed convergence rates numerically in the previous notebook, we focus in this notebook on the numerical analysis and presenting ideas on how to prove the observed convergence rates.

The notebook is based on the following works: Gawlik. High-Order Approximation of Gaussian Curvature with Regge Finite Elements, SIAM Journal on Numerical Analysis (2020)., Berchenko-Kogan, Gawlik. Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions, Found Comput Math (2022), Gopalakrishnan, Neunteufel, Schöberl, Wardetzky. Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics, The SMAI Journal of computational mathematics (2023)., and Gopalakrishnan, Neunteufel, Schöberl, Wardetzky. On the improved convergence of lifted distributional Gauss curvature from Regge elements, Results in Applied Mathematics (2024)..

The distributional Gauss curvature

\[\begin{align*} \widetilde{K\,\omega}(\varphi) = \sum_{T\in\mathcal{T}}\int_TK|_T\,\varphi\,\omega|_T+\sum_{E\in\mathcal{E}}\int_E[\![\kappa_g]\!]\,\varphi\,\omega_E+\sum_{V\in\mathcal{V}}\sphericalangle_V\,\varphi(V) \end{align*}\]

is highly nonlinear in the metric, making numerical analysis unfeasible in its current form. Fortunately, the distributional Gauss curvature entails an integral representation generalizing the Hellan-Herrmann-Johnson method from the Euclidean to the covariant setting.

Integral representation of distributional Gauss curvature

Christiansen showed in Christiansen. On the linearization of Regge calculus. Numer. Math. (2011). that the linearization of the Einstein-Hilbert integral around the Euclidean metric is the incompatibility operator in 3D. Gawlik investigated in Gawlik. High-Order Approximation of Gaussian Curvature with Regge Finite Elements, SIAM Journal on Numerical Analysis (2020). the variation of the densitized Gauss curvature around a given general metric. There holds for a smooth metric the following Gateaux derivative in a direction \(\sigma\in C^{\infty}(\Omega,\mathbb{R}^{2\times 2}_{\mathrm{sym}})\)

\[\begin{align*} D_g(K(g)\omega(g))[\sigma]:= \lim\limits_{t\to 0}\frac{K(g+t\sigma)\omega(g+t\sigma)-K(g)\omega(g)}{t} = \frac{1}{2}\mathrm{div}_g\mathrm{div}_g(\mathbb{S}_{g}\sigma)\omega(g), \end{align*}\]

where \(\mathbb{S}_g\sigma = \sigma -\mathrm{tr}_g(\sigma)g\) and subscript \(g\) indicates that covariant differential operators are considered. There also holds the following identity with the incompatibility operator

\[\begin{align*} \mathrm{div}_g\mathrm{div}_g(\mathbb{S}_{g}\sigma)\omega(g) = -\mathrm{inc}_g(\sigma)\omega(g),\qquad \mathrm{inc}_g:=\mathrm{curl}_g\mathrm{curl}_g. \end{align*}\]

In the Euclidean case, it would become \(\mathrm{div}\mathrm{div}(\mathbb{S}\sigma)\). Let \(n\) be the outward unit normal to each element and \(t\) a unit tangent to its edges. Using \(\sigma_h\in \mathrm{Reg}_h^k\) in the Regge space, \(\mathbb{S}\sigma_h\) becomes normal-normal continuous, fitting the Hellan-Herrmann-Johnson method Comodi. The Hellan-Herrmann-Johnson method: some new error estimates and postprocessing. Math. Comp. (1989). For \(u_h\in\mathrm{Lag}^{k+1}\), the corresponding bilinear form is

\[\begin{split}\begin{align*} b_h(\delta;\sigma_h,u_h) &= \sum_{T\in\mathcal{T}}\left(\int_T \mathbb{S}\sigma_h : \nabla^2u_h - \int_{\partial T}(\mathbb{S}\sigma_h)_{nn}\partial_nu_h\right)\\ &= \sum_{T\in\mathcal{T}}\left(\int_T \mathbb{S}\sigma_h : \nabla^2u_h + \int_{\partial T}\sigma_h(t,t)\partial_nu_h\right), \end{align*}\end{split}\]

The plus sign follows from \((\mathbb{S}\sigma_h)_{nn}=\sigma_h(n,n)-\mathrm{tr}(\sigma_h)=-\sigma_h(t,t)\) in two dimensions. The edge terms use the outward normal derivatives from the two adjacent elements.

Gawlik recognized that the HHJ method could be extended to the covariant case. For \(g\in \mathrm{Reg}_h^k\), with \(g\)-unit outward normal \(n\) and tangent \(t\), the corresponding form is

\[\begin{split}\begin{align*} b_h(g;\sigma_h,u_h) &= \sum_{T\in\mathcal{T}}\left(\int_T g(\mathbb{S}_g\sigma_h,\nabla_g^2u_h)\,\omega|_T(g)-\int_{\partial T}(\mathbb{S}_g\sigma_h)_{nn}\partial_nu_h\,\omega_{\partial T}(g)\right)\\ &= \sum_{T\in\mathcal{T}}\left(\int_T g(\mathbb{S}_g\sigma_h,\nabla_g^2u_h)\,\omega|_T(g)+\int_{\partial T}\sigma_h(t,t)\partial_nu_h\,\omega_{\partial T}(g)\right). \end{align*}\end{split}\]

The variations of the jump of the geodesic curvature and the angle defect Berchenko-Kogan, Gawlik. Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions, Found Comput Math (2022) combine with the element curvature variation to give this covariant HHJ form. With the outward-normal convention, the geodesic-curvature variation has a minus sign:

\[\begin{align*} D_g(\kappa_g\,\omega_E)[\sigma]= -\frac{1}{2}\left((\mathrm{div}_g(\mathbb{S}_g\sigma))(n)+\partial_t(\sigma(n,t))\right)\,\omega_E. \end{align*}\]

proving that

\[\begin{align*} D_g \widetilde{K\omega}(u_h)[\sigma_h]=\frac{1}{2}b_h(g;\sigma_h,u_h), \end{align*}\]

Noting that the Gauss curvature for the Euclidean metric is zero, we obtain with the fundamental theorem of calculus the following integral representation of the Gauss curvature

\[\begin{align*} \widetilde{K\omega}(u_h)= \frac{1}{2}\int_0^1b_h(g(t);\sigma,u_h)\,dt,\qquad g(t) = \delta+t(g_h-\delta),\quad \sigma=\dot{g}(t) = g_h-\delta, \end{align*}\]

where \(\delta\) denotes the Euclidean metric. By consistency, the exact Gauss curvature from the smooth, exact metric also fulfils the integral representation. Thus, the error has the following integral representation

\[\begin{align*} \langle \widetilde{K\omega}(g_h)-K\omega,u_h\rangle = \frac{1}{2}\int_0^1b_h(g(t);\sigma,u_h)\,dt,\qquad g(t) = g+t(g_h-g),\quad \sigma=\dot{g}(t) = g_h-g \end{align*}\]

The integrand is linear enough to perform numerical analysis to prove the observed convergence rates. For the details of the (technical) proofs, we refer to the above-mentioned literature.

Improved convergence rates by the canonical Regge interpolant

When using the canonical Regge interpolation operator to approximate the metric tensor, i.e. \(g_h=\mathcal{I}^{\mathrm{Reg}^k}g\), one can achieve one or even two additional orders of convergence. For \(g_h=\mathcal{I}^{\mathrm{Reg}^k}g\) and \(K_h\in\mathrm{Lag}^{k+1}\) the following convergence rates hold

\[\begin{align*} \|K_h-K\|_{L^2}\le C h^{k},\qquad \|K_h-K\|_{H^{-1}}\le C h^{k+1}. \end{align*}\]

Using stronger norms is also possible and reduces the rates as usual. The proof’s strategy relies on exploiting the definition of the canonical Regge interpolant that the first \(k+1\) tangential-tangential moments at the edges (starting with the zeroth-order moment) and, for higher order, some internal moments of \(g\) and \(g_h\) coincide. This allows the extraction of one additional order of convergence. The technical proof can be found in Gopalakrishnan, Neunteufel, Schöberl, Wardetzky. Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics, The SMAI Journal of Computational Mathematics (2023)..

Reducing the polynomial order of \(K_h\) to coincide with the metric’s order, i.e. \(K_h\in \mathrm{Lag}^k\) and \(g_h=\mathcal{I}^{\mathrm{Reg}^k}g\), yields another improvement for \(k\ge 1\):

\[\begin{align*} \|K_h-K\|_{L^2}\le C h^{k+1},\qquad \|K_h-K\|_{H^{-1}}\le C h^{k+2},\qquad k\ge 1. \end{align*}\]

In fact, one obtains the optimal convergence possible in the \(L^2\)-norm. Again, the key ingredient is the definition of the canonical Regge interpolant. The details can be found in Gopalakrishnan, Neunteufel, Schöberl, Wardetzky. On the improved convergence of lifted distributional Gauss curvature from Regge elements, Results in Applied Mathematics (2024).. Note that the convergence rates highly depend on the chosen polynomial approximation spaces for \(g_h\) and \(K_h\), and whether the canonical Regge interpolation operator is used or something else.