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

Ziad Aldirany (1) , Régis Cottereau (2) , Marc Laforest (1) , Serge Prudhomme (1)

1 EPM - École Polytechnique de Montréal
2 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]

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 is shown to be of order with respect to the element size h and of order with respect to the degree p, where q is the spatial regularity of the solution. These results improve on past estimates in the 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.

Mots clés: finite element method; spectral element method; wave equation; multi-dimensional; convergence; error estimation; explicit; leap-frog scheme

Domaines: Analyse numérique [math.NA]

Ziad Aldirany, Régis Cottereau, Marc Laforest, Serge Prudhomme. Optimal error analysis of the spectral element method for the 2D homogeneous wave equation. Computers & Mathematics with Applications, 2022, 119, pp.241-256. ⟨10.1016/j.camwa.2022.05.038⟩. ⟨hal-03766726v3⟩