Numerical Approximation of Biharmonic Wave Maps into Spheres

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1160-1182, June 2025.
Abstract. We construct a structure preserving nonconforming finite element approximation scheme for the biharmonic wave maps into spheres equations. It satisfies a discrete energy law and preserves the nonconvex 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 (nonconforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension [math]. The convergence analysis in dimensions [math] is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the nonconforming 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.