Analysis of the Crouzeix-Raviart Surface Finite Element Method for vector-valued Laplacians

TL;DR

The paper addresses discretizing the vector-valued Bochner Laplace equation on smooth surfaces using a nonconforming surface Crouzeix–Raviart finite element on a polygonal approximation $\Gamma_h$. It develops interpolation estimates for $\Pi_h^{\mathrm{tan}}$ and proves optimal $H^1$- and $L^2$-error bounds, including a geometric error analysis and a duality-based $L^2$-estimate, establishing bounds such as $\|\tilde{\mathbf{u}}-\mathbf{u}_h\|_h \le c h \|\mathbf{f}\|_{L^2(\Gamma)}$ and $\|\mathbf{P}_h(\tilde{\mathbf{u}}-\mathbf{u}_h)\|_{L^2(\Gamma_h)} \le c h^2 \|\mathbf{f}\|_{L^2(\Gamma)}$. Numerical experiments on the unit sphere and a torus validate the predicted rates, confirming quadratic convergence in $L^2$ and linear convergence in $H^1$. The work delivers a penalty-free, tangential, nonconforming discretization for surface vector Laplacians, with potential impact on climate-model surface flows and other geophysical applications. It combines a rigorous geometric analysis with a robust nonconforming framework to achieve efficient and accurate surface discretizations without additional geometric penalties.

Abstract

Recently, a nonconforming surface finite element was developed to discretize 3d vector-valued compressible flow problems arising in climate modeling. In this contribution we derive an error analysis for this approach on a vector-valued Laplace problem, which is an important operator for fluid-equations on the surface. In our setup, the problem is approximated via edge-integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The latter is continuous at the midpoints of the edges in each vector component. This setup is numerically efficient and straightforward to implement. For this Crouzeix-Raviart discretization we introduce interpolation estimates, derive optimal error bounds in the H1-norm and L2-norm and present an estimate for the geometric error. Numerical experiments validate the theoretical results.

Figures (4)

• Figure 1: a): The tangential plane which coincides with the sphere at the vertices of the triangle. The outward pointing normal vector (blue) is named $\mathbf{n}_h$. The tangential and conormal of the flat triangle $K$ (shaded in gray) to an edge $E$ is called $\mathbf{n}_E, \boldsymbol{\tau}_E$. Accordingly the tangential and conormal of the curved triangle $K^l$ (orange) is named $\mathbf{n}_E^l, \boldsymbol{\tau}_E^l$. b): Crouzeix-Raviart basis function. c): The outward pointing conormal vectors $\mathbf{n}_E^\pm$ that share an edge $E$.
• Figure 2: Sphere: a): Grid after three refinements. b): First component of the solution $\mathbf{u}_{\Gamma_S}$ with vorticity streamlines.
• Figure 3: Torus: a): Grid after three refinements. b): First component of the solution $\mathbf{u}_{\Gamma_T}$ with vorticity streamlines.
• Figure 4: Errors in the norms $\| \tilde{\mathbf{u}} -\mathbf{u}_h\|_{H^1(\Gamma_h)}$ and $\|\mathbf{P}_h(\tilde{\mathbf{u}} -\mathbf{u}_h)\|_{L^2(\Gamma_h)}$ on the sphere and on the tours. The absolute values are given in Table \ref{['tab:sphere']} and Table \ref{['tab:torus']}, respectively.

Theorems & Definitions (25)

• Lemma 3.1: Geometry approximation
• proof
• Lemma 3.2: Norm equivalence
• proof
• Lemma 5.1: Interpolation Estimates
• Remark 5.2
• proof
• Theorem 5.3: Strang's second Lemma
• Lemma 5.4: Nonconformity Error Estimate
• proof