SUMMARYWe consider isotropic elastic wave propagation with continuous mass-lumped finite elements on tetrahedra with explicit time stepping. These elements require higher-order polynomials in their interior to preserve accuracy after mass lumping and are only known up to degree 3. Global assembly of the symmetric stiffness matrix is a natural approach but requires large memory.Local assembly on the fly, in the form of matrix-vector products per element at each time step, has a much smaller memory footprint. With dedicated expressions for local assembly, our code ran about 1.3 times faster for degree 2 and 1.9 times for degree 3 on a simple homogeneous test problem, using 24 cores. This is similar to the acoustic case. For a more realistic problem, the gain in efficiency was a factor 2.5 for degree 2 and 3 for degree 3. For the lowest degree, the linear element, the expressions for both the global and local assembly can be further simplified.In that case, global assembly is more efficient than local assembly. Among the three degrees, the element of degree 3 is the most efficient in terms of accuracy at a given cost.

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 and 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, 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 show have a small amplitude when the true eigenfunction is projected onto them. We analyse 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 constant.

SUMMARYThe second-order formulation of the wave equation is often used for spectral-element discretizations. For some applications, however, a first-order formulation may be desirable. It can, in theory, provide much better accuracy in terms of numerical dispersion if the consistent mass matrix is used and the degree of the polynomial basis functions is odd. However, we find in the 1-D case that the eigenvector errors for elements of degree higher than one are larger for the first-order than for the second-order formulation. These errors measure the unwanted cross talk between the different eigenmodes. Since they are absent for the lowest degree, that linear element may perform better in the first-order formulation if the consistent mass matrix is inverted. The latter may be avoided by using one or two defect-correction iterations. Numerical experiments on triangles confirm the superior accuracy of the first-order formulation. However, with a delta-function point source, a large amount of numerical noise is generated. Although this can be avoided by a smoother source representation, its higher cost and the increased susceptibility to numerical noise make the second-order formulation more attractive.

SUMMARYWe 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 LegendreGauss-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.

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