Spectral elements with mass lumping allow for explicit time stepping and are therefore attractive for modelling seismic wave propagation. Their formulation on rectangular elements is straighforward, but for tetrahedra only elements up to degree 3 are known. To preserve accuracy after mass lumping, these elements require additional nodes that make them computationally more expensive. Here, we propose a new, less restrictive accuracy condition for the construction of these continuous mass-lumped elements. This enables us to construct several new tetrahedral elements. The new degree-2 and degree-3 elements require 15 and 32 nodes, while the existing ones have 23 and 50 nodes per element, respectively. We also developed degree-4 tetrahedral elements with 60, 61, or 65 nodes per element. Numerical examples confirm that the various mass-lumped elements maintain the optimal order of accuracy and show that the new elements are significantly more efficient in terms of accuracy versus compute time than the existing ones.@en

We present a new accuracy condition for constructing mass-lumped finite elements. This new condition is less restrictive than the one that has been used for several decades and enabled us to construct new mass-lumped tetrahedral elements of degrees 2,3, and 4. The new degree-2 and degree-3 elements require 14 and 32 nodes, respectively, while the degree-2 and degree-3 elements currently available in the literature require 23 and 50 nodes, respectively. Mass-lumped tetrahedral elements of degree 4 had not been found until now. The resulting mass-lumped finite element method is suitable for 3D wave propagation problems, since it can accurately capture the effects of a complex geometry and since it results in a fully explicit time-stepping scheme. A dispersion analysis and several numerical tests illustrates the efficiency of this new method. In particular, they illustrate a significant reduction in degrees of freedom, number of time steps, and computation time for a given accuracy compared to other finite element methods, such as the previous mass-lumped finite element methods or the discontinuous Galerkin method.@en