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Numerical approximation of bi-harmonic wave maps into spheres

Ľubomír Baňas, Sebastian Herr

TL;DR

This work tackles the numerical approximation of bi-harmonic wave maps into the sphere by developing a structure-preserving, non-conforming finite element scheme that enforces the sphere constraint at mesh nodes via a discrete Lagrange multiplier and satisfies a discrete energy law. Convergence is proven in one spatial dimension, while a stabilization strategy is introduced to achieve convergence in higher dimensions, along with a conservative energy-preserving variant. The analysis combines a discrete product rule, compactness arguments, and energy estimates to bridge the gap between discrete and weak formulations. Numerical experiments illustrate the scheme's stability, the regularizing effect of the bi-Laplacian, and the practical feasibility of the approach across dimensions.

Abstract

We construct a structure preserving non-conforming finite element approximation scheme for the bi-harmonic wave maps into spheres equation. It satisfies a discrete energy law and preserves the non-convex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points via a discrete Lagrange multiplier. This approach restricts the spatial approximation to the (non-conforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension $d=1$. The convergence analysis in dimensions $d>1$ is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the non-conforming setting. Hence, we show convergence of the numerical approximation in higher-dimensions by introducing additional stabilization terms in the numerical approximation. We present numerical experiments to demonstrate the performance of the proposed numerical approximation and to illustrate the regularizing effect of the bi-Laplacian which prevents the formation of singularities.

Numerical approximation of bi-harmonic wave maps into spheres

TL;DR

This work tackles the numerical approximation of bi-harmonic wave maps into the sphere by developing a structure-preserving, non-conforming finite element scheme that enforces the sphere constraint at mesh nodes via a discrete Lagrange multiplier and satisfies a discrete energy law. Convergence is proven in one spatial dimension, while a stabilization strategy is introduced to achieve convergence in higher dimensions, along with a conservative energy-preserving variant. The analysis combines a discrete product rule, compactness arguments, and energy estimates to bridge the gap between discrete and weak formulations. Numerical experiments illustrate the scheme's stability, the regularizing effect of the bi-Laplacian, and the practical feasibility of the approach across dimensions.

Abstract

We construct a structure preserving non-conforming finite element approximation scheme for the bi-harmonic wave maps into spheres equation. It satisfies a discrete energy law and preserves the non-convex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points via a discrete Lagrange multiplier. This approach restricts the spatial approximation to the (non-conforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension . The convergence analysis in dimensions is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the non-conforming setting. Hence, we show convergence of the numerical approximation in higher-dimensions by introducing additional stabilization terms in the numerical approximation. We present numerical experiments to demonstrate the performance of the proposed numerical approximation and to illustrate the regularizing effect of the bi-Laplacian which prevents the formation of singularities.
Paper Structure (12 sections, 6 theorems, 97 equations, 5 figures)

This paper contains 12 sections, 6 theorems, 97 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Weak solution
  4. Numerical approximation
  5. Convergence of the numerical approximation
  6. Higher-dimensional case
  7. Existence
  8. Convergence
  9. Conservative scheme
  10. Numerical experiments
  11. Solution of the nonlinear systems
  12. Numerical experiments

Key Result

Lemma 3.1

Let $\mathcal{T}_h$ be a quasiuniform triangulation of $\Omega \subset \mathbb{R}^d$, and $(\, {\bf U}^0, {\bf V}^0\, ) \in {\bf V}_h \times {\bf V}_h$ such that $\vert {\bf U}^0({\bf z}) \vert = 1$, and $\bigl\langle {\bf U}^0({\bf z}), {\bf V}^{0}({\bf z})\bigr\rangle = 0$ for all ${\bf z} \in \ma

Figures (5)

  • Figure 1: Evolution of the discrete energy (left) and of the gradient $\|\nabla \mathbf{U}^n\|_{\mathbf{L}^\infty}$ (right) for $h=2^{k}$, $k=5,6,7,8$.
  • Figure 2: Numerical solution computed with $h=2^{-5}$ at time $t=0,0.02,0.045,0.095,0.13,0.24$ (from left to right, top to bottom).
  • Figure 3: Evolution of the respective discrete energies $\mathcal{E}_h$ and $\mathcal{E}_{h, {\tt stab}}$ (left) and of $\frac{C_{\tt stab} h^2}{2}\|\nabla \mathbf{W}^n\|^2$ (right) for $h=2^{k}$, $k=5,6$.
  • Figure 4: Evolution of the respective discrete energies $\mathcal{E}_h$ and $\mathcal{E}_{h, {\tt stab}}$ (left) and of $\frac{C_{\tt stab} h^2}{2}\|\nabla \mathbf{W}^n\|^2$ (middle) and the evolution of $\|\nabla \mathbf{U}^n\|_{\mathbf{L}^\infty}$ (right) for $h=2^{k}$, $k=5,6,7$.
  • Figure 5: Numerical solution computed with $h=2^{-5}$ (without stabilization) at time $t=0,0.005,0.05,0.075,0.09,0.095$ (from left to right, top to bottom).

Theorems & Definitions (18)

  • Remark 2.1
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 8 more