Numerical modeling of the scalar and elastic wave equations with Chebyshev spectral finite element

One and two-dimensional finite elements are formulated with Chebyshev polynomial based shape functions for use in solutions to the scalar and elastic wave equations. The accuracy of these elements with three mass matrix formulations is compared to that of lower order p-elements. Chebyshev finite element solutions with either consistent or row-summed mass matrices are shown to exhibit substantially lower dispersive and natural frequency errors than those of lower order p-elements. These formulations generally display increasing accuracy with: (1) increasing shape function order, (2) increasing mesh refinement and (3) decreasing time step. Additionally, the accuracies of the explicit row-summed mass matrix solutions are equivalent to or better than those employing consistent mass matrices. All temporal discretizations use a central-difference-in-time formulation.;Computational costs for prescribed levels of solution dispersive error are studied for Chebyshev spectral and p-elements. The Chebyshev spectral finite elements offer significant computational cost savings over the family of p-elements. Sample problems highlight the fidelity of the Chebyshev spectral finite element solutions to the wave equation at modest levels of mesh refinement.