Convergent discretization of heat and wave map flows to spheres using approximate discrete Lagrange multipliers

We propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres. The finite-element based schemes preserve a unit length constraint at the nodes by means of approximate discrete Lagrange multipliers, satisfy a discrete energy law, and iterates are shown to converge to weak solutions of the continuous problem. Comparative computational studies are included to motivate finite-time blow-up behavior in both cases.