Implicit spectral element methods and Neumann–Neumann preconditioners for acoustic waves

A numerical approximation of the acoustic wave equation with first order absorbing boundary conditions is considered. The discretization is based on conforming spectral elements in space and implicit finite differences in time. A stability analysis based on the energy method is developed for the fully discrete scheme. The linear system arising at each step is solved by the conjugate gradient method with Balancing Neumann–Neumann preconditioning. Several numerical results illustrate the stability and convergence properties of the approximation schemes, that result spectrally accurate in space and up to second-order in time, while the Neumann–Neumann solver at each time step is scalable and quasi-optimal.