Dispersion Analysis of Finite-element Schemes for a First-order Formulation of the Wave Equation

We investigated one-dimensional numerical dispersion curves and error behaviour of four finite-element schemes with polynomial basis functions: the standard elements with equidistant nodes, the Legendre-Gauss-Lobatto points, the Chebyshev-Gauss-Lobatto nodes without a weighting function and with. Mass lumping, required for efficiency reasons and enabling explicit time stepping, may adversely affect the numerical error. We show that in some cases, the accuracy can be improved by applying one iteration on the full mass matrix, preconditioned by its lumped version. For polynomials of degree one, this improves the accuracy from second to fourth order in the element size. In other cases, the improvement in accuracy is less dramatic.