Improving the accuracy of mass-lumped finite-elements in the first-order formulation of the wave equation by defect correction

Finite-element discretizations of the acoustic wave equation in the time domain often employ mass lumping to avoid the cost of inverting a large sparse mass matrix. For the second-order formulation of the wave equation, mass lumping on Legendre–Gauss–Lobatto points does not harm the accuracy. Here, we consider a first-order formulation of the wave equation. In that case, the numerical dispersion for odd-degree polynomials exhibits super-convergence with a consistent mass matrix but mass lumping destroys that property. We consider defect correction as a means to restore the accuracy, in which the consistent mass matrix is approximately inverted using the lumped one as preconditioner. For the lowest-degree element on a uniform mesh, fourth-order accuracy in 1D can be obtained with just a single iteration of defect correction.

The numerical dispersion curve describes the error in the eigenvalues of the discrete set of equations. However, the error in the eigenvectors also play a role, in two ways. For polynomial degrees above one and when considering a 1-D mesh with constant element size and constant material properties, a number of modes, equal to the maximum polynomial degree, are coupled. One of these is the correct physical mode that should approximate the true eigenfunction of the operator, the other are spurious and should have a small amplitude when the true eigenfunction is projected onto them. We analyze the behaviour of this error as a function of the normalized wavenumber in the form of the leading terms in its series expansion and find that this error exceeds the dispersion error, except for the lowest degree where the eigenvector error is zero. Numerical 1-D tests confirm this behaviour.

We briefly analyze the 2-D case, where the lowest-degree polynomial also appears to provide fourth-order accuracy with defect correction, if the grid of squares or triangles is highly regular and material properties are constant.