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

Title

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

 Author

Mulder, W.A. (TU Delft Applied Geophysics and Petrophysics; Shell Global Solutions International B.V.) ORCID 0000-0001-7020-9297
Shamasundar, R. (TU Delft Applied Geophysics and Petrophysics)

Date

2024

 Abstract

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 timeand frequency-domain are included.

 To reference this document use:

http://resolver.tudelft.nl/uuid:1c281ed2-d91b-41bc-a903-28079437ddf5

 DOI

https://doi.org/10.2528/PIERM24011601

 Embargo date

2025-08-23

 ISSN

1937-8726

 Source

Progress in Electromagnetics Research (PIER) M, 126, 147-155

 Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

 Part of collection

Institutional Repository

 Document type

journal article

 Rights

© 2024 W.A. Mulder, R. Shamasundar
Files
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