Convergence analysis of energy conserving explicit local time-stepping methods for the wave equation
For computational scientists simulating wave phenomena, this work offers a theoretically grounded method to accelerate simulations on locally refined meshes, though the proof is for a specific existing method.
The paper provides a rigorous convergence proof for an explicit local time-stepping method combined with finite elements for the wave equation, overcoming the time-step bottleneck caused by locally refined meshes. Numerical results demonstrate the method's effectiveness for corner singularities.
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 [18] 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 a rigorous convergence proof is presented 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.