2024/09/17 by Ľubomír Baňas, Sebastian Herr, Baňas, Ľubomír +1
Computer Science · Earth and Planetary Sciences · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.2409.11366
openalex publication_date 2024/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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.