Dispersion and Eigenvector Error Analysis of Simplicial Cubic Hermite Elements for 1-D and 2-D Wave Propagation Problems

Dispersion error analysis can help to assess the performance of finite-element discretizations of the wave equation. Although less general than the convergence estimates offered by standard finite-element error analysis, it can provide more detailed insight as well as practical guidelines in terms of the number of elements per wavelength needed for acceptable results. We present eigenvalue and eigenvector error estimates for cubic Hermite elements on an equidistant 1-D mesh and on a regular structured 2-D triangular mesh consisting of squares cut in half. The results show that in 1D, the spectrum consists of 2 modes. If these are unwrapped, the spectrum is effectively doubled. The eigenvalue or dispersion error stays below 7% across the entire spectrum. The error in the corresponding eigenvectors, however, increases rapidly once the number of elements per wavelength decreases to one. In terms of element size, the dispersion error is of order 6 and the eigenvector error of order 4. The latter is consistent with the classic finite-element error estimate. In 2D, we provide eigenvalue and eigenvector errors as a series expansion in the element size and obtain the same orders. 2-D numerical tests in the time- and frequency-domain are included.