A note on the rate of convergence of integration schemes for closed surfaces

Abstract

In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree polynomials. Additionally, we provide some numerical examples that illustrate our findings and propose a potential approach that overcomes the problems associated with the original one.

Figures (7)

• Figure 1: Representation of a smooth surface parametrization, where every region $V_{i}$ forms a curved triangle.
• Figure 2: Construction of the second order approximation of the smooth surface $\mathcal{M}^{2}_{h}$ (blue line). A simplex of a ‘base’ triangulation $\mathcal{M}_{h}$ (green line) is shown. The interpolation nodes, here the center $\bar{q_{i}}$ of an edge, are projected (grey line) onto the smooth surface $\mathcal{M}$ (red line) via the projection $\pi$. The projected nodal points $\pi\left(\bar{q_{i}}\right)$ and the vertices of $\mathcal{M}_{h}$ are then interpolated, giving the second order approximation of the smooth surface $\mathcal{M}^{2}_{h}$.
• Figure 3: Lagrange parametrization for a torus and a sphere using equidistant nodes on vertices and edges.
• Figure 4: Illustration of bisection refinement and a duo of symmetric triangles obtained from the triangulation $\mathcal{T}^{1}_{h}$.
• Figure 5: Relative errors by integrating the Gaussian curvature over the torus with radii $R=2,\,r=1$ with the ideal convergence lines $h^{n}$.

Theorems & Definitions (14)

• Definition 3.1: symmetric triangles
• Remark 3.2: symmetric triangulation
• Theorem 3.3
• Lemma 3.4
• proof
• Lemma 3.5: Chien1993
• Lemma 3.6
• proof
• proof : Poof of theorem \ref{['main.thm']}
• Theorem 4.1