Finite Element Approximations of Wave Maps into Spheres

Abstract

Three fully discrete finite element methods are developed for approximating wave maps into the sphere based on two different approaches. The first method is an explicit scheme and the numerical solution satisfies the sphere-constraint exactly at every node. The second and third methods are implicit schemes which are based on a penalization approach, their numerical solutions satisfy the sphere-constraint approximately, and the quality of approximations is controlled by a small penalization parameter. Discrete energy conservation laws which mimic the underlying differential conservation law are established, and convergence of all proposed methods is proved. Computational experiments are also provided to validate the proposed methods and to present numerical evidence for possible finite-time blow-ups of the wave maps.