Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time step everywhere with a crippling effect on any explicit time-marching method. In [J. Diaz and M. J. Grote, SIAM J. Sci. Comput., 31 (2009), pp. 1985--2014] a leap-frog (LF)-based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time steps in the locally refined region and larger steps elsewhere. Here optimal convergence rates are rigorously proved for the fully discrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.