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 spectral element method. To be precise, method studied here takes advantage of Gauss-Lobatto-Legendre quadrature, thus resulting in under-integrated elements but diagonal mass matrix. The approximation H1 is shown order O(hp) with respect size h O(p−q) degree p, where q spatial regularity solution. These results improve on past estimates L2 norm, particularly h. Specific assumptions discretization use triangulation quadrilaterals constructed via affine transformations from reference square second time leap-frog scheme.