Numerical approximation of bi-harmonic wave maps into spheres

TL;DR

This paper develops a structure-preserving finite element scheme for bi-harmonic wave maps into spheres, ensuring energy conservation and sphere constraint preservation, with proven convergence in 1D and extended results in higher dimensions.

Contribution

It introduces a novel non-conforming finite element approximation that maintains key geometric properties and demonstrates convergence, including stabilization techniques for higher dimensions.

Findings

The scheme preserves discrete energy and sphere constraints.

Convergence is proven in 1D; higher dimensions require stabilization.

Numerical experiments show the regularizing effect of the bi-Laplacian.

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.