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Convergence analysis of the intrinsic surface finite element method

Elena Bachini, Mario Putti

TL;DR

This work complements the original derivation of ISFEM with its complete convergence theory and proposes the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities.

Abstract

The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and makes direct use of the resulting covariant basis. Starting from a shape-regular triangulation of the surface, existence of a local parametrization for each triangle is exploited to approximate relevant quantities on the local chart. Standard two-dimensional FEM techniques in combination with surface quadrature rules complete the ISFEM formulation thus achieving a method that is fully intrinsic to the surface and makes limited use of the surface embedding only for the definition of basis functions. However, theoretical properties have not yet been proved. In this work we complement the original derivation of ISFEM with its complete convergence theory and propose the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities. Numerical experiments are included to support the theoretical results.

Convergence analysis of the intrinsic surface finite element method

TL;DR

This work complements the original derivation of ISFEM with its complete convergence theory and proposes the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities.

Abstract

The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and makes direct use of the resulting covariant basis. Starting from a shape-regular triangulation of the surface, existence of a local parametrization for each triangle is exploited to approximate relevant quantities on the local chart. Standard two-dimensional FEM techniques in combination with surface quadrature rules complete the ISFEM formulation thus achieving a method that is fully intrinsic to the surface and makes limited use of the surface embedding only for the definition of basis functions. However, theoretical properties have not yet been proved. In this work we complement the original derivation of ISFEM with its complete convergence theory and propose the analysis of the stability and error estimates by carefully tracking the role of the geometric quantities in the constants of the error inequalities. Numerical experiments are included to support the theoretical results.
Paper Structure (21 sections, 14 theorems, 68 equations, 3 figures)

This paper contains 21 sections, 14 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Intrinsic Surface FEM
  3. Geometrical setting
  4. Differential operators
  5. Intrinsic variational formulation
  6. Function spaces
  7. Intrinsic variational problem
  8. Poincaré inequality
  9. Well-posedness
  10. Intrinsic finite element method
  11. The surface triangulation
  12. Intrinsic spatial discretization
  13. Surface quadrature rules
  14. Discrete norms
  15. ISFEM formulation
  16. ...and 6 more sections

Key Result

Lemma 2.2.2

Let $\Gamma$ a $C^{0}$-regular surface without boundary and let $u\in H^{1} ( \Gamma )$ be a function with average given by $\bar{u}=\frac{1}{ \left| \Gamma \right| }\int_{ \Gamma }u$. Then, there exists a constant $C_{_{ \Gamma }}>0$ such that:

Figures (3)

  • Figure 1: Surfaces described by \ref{['eq:surface-trig']} with $r=2$, $k=5$ and $a=0, 0.5, 2$, respectively in the left, middle and right panels. The color map shows the value of $\sqrt{\operatorname{det}( \mathcal{G} ( \Gamma ))}$.
  • Figure 2: TC1: Numerical convergence of $L^{2}$-errors for the solution (left) and its gradient (right) vs $h$ on the surface triangulation. The convergence lines are obtained by means of least-square approximation considering the last 2 point values. The different lines denote the three different values of $a$ considered: solid line with triangular data points is used for the case $a=0$, dashed line with circular data points for $a=0.5$, and dotted line with diamond data points for the case $a=2$. The optimal theoretical slope is represented by the lower right triangles.
  • Figure 3: TC2: Numerical solution on the sphere (mesh level $\ell=1$), left panel, and numerical convergences in the $L2-$norm for both the solution (solid line with circular data points) and its gradient (dotted line with diamond data points), right panel. The convergence lines are obtained by approximating via least-square the last 3 point values.

Theorems & Definitions (27)

  • Remark 2.1.1
  • Definition 2.1.2
  • Lemma 2.2.2
  • Corollary 2.2.3
  • Lemma 2.2.4
  • Lemma 2.2.5
  • proof
  • Lemma 2.2.6: Lax-Milgram theorem
  • Remark 2.3.1
  • Remark 2.3.2
  • ...and 17 more