More Continuous Mass-Lumped Triangular Finite Elements

When solving the wave equation with finite elements, mass lumping allows for explicit time stepping, avoiding the cost of a lower-upper decomposition of the large sparse mass matrix. Mass lumping on the reference element amounts to numerical quadrature. The weights should be positive for stable time stepping and preserve numerical accuracy. The standard triangular polynomial elements, except for the linear element, do not have these properties. Accuracy can be preserved by augmenting them with higher-degree polynomials in the interior. This leaves the search for elements with positive weights, which were found up to degree 9 by various authors. The classic accuracy condition, however, is too restrictive. A sharper, less restrictive condition recently led to new mass-lumped tetrahedral elements up to degree 4. Compared to the known ones up to degree 3, they have less nodes and are computationally more efficient. The same criterion is applied here to the construction of triangular elements. For degrees 2 to 4, these turn out to be identical to the known ones. For degree 5, the number of nodes is the same as for the known element, but now there are infinitely many solutions. Some of these have a considerably larger stability limit for time stepping. For degree 6, two elements are found with less nodes than the known ones. For degree 7, one element with less nodes was found but with a negative weight, making it useless for time stepping with the wave equation. If the number of nodes is the same as for the classic element, there are now infinitely many solutions. Numerical tests for a homogeneous wave-propagation problem with a point source confirm the expected accuracy of the new elements. Some of them require less compute time than those obtained with the more restrictive accuracy criterion.