Optimal error analysis of the spectral element method for the 2D homogeneous wave equation

Optimal a priori error bounds are theoretically derived, and numerically verified, for approximate solutions to the 2D homogeneous wave equation obtained by the spectral element method. To be precise, the spectral element method studied here takes advantage of the Gauss-Lobatto-Legendre quadrature, thus resulting in under-integrated elements but a diagonal mass matrix. The approximation error in H1 is shown to be of order O(h(p)) with respect to the element size h and of order O(p(-q)) with respect to the degree p, where q is the spatial regularity of the solution. These results improve on past estimates in the L-2 norm, particularly with respect to h. Specific assumptions on the discretization by the spectral element method are the use of a triangulation by quadrilaterals constructed via affine transformations from a reference square element and of a second order discretization in time by the leap-frog scheme.