Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique
Abstract
Purpose
The purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and fourth-order Sturm–Liouville (SL) problems with variable coefficients.
Design/methodology/approach
After the conventional FE solution for an eigenpair (i.e. eigenvalue and eigenfunction) of a particular order has been obtained on a given mesh, a novel strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the element energy projection (EEP) technique for linear problems to calculate the super-convergent FE solutions for eigenfunctions anywhere on any element. These EEP super-convergent solutions are used to estimate the FE solution errors and to guide mesh refinements, until the accuracy matches user-preset error tolerance on both eigenvalues and eigenfunctions.
Findings
Numerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.
Research limitations/implications
The method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way.
Originality/value
Comprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.
Keywords
Acknowledgements
The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant nos. 51378293, 50678093, 51078199, 51078198), the Chinese Ministry of Education (Grant no. IRT00736) and the Cardiff Advanced Chinese Engineering Centre.
Citation
Yuan, S., Ye, K., Wang, Y., Kennedy, D. and Williams, F.W. (2017), "Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique", Engineering Computations, Vol. 34 No. 8, pp. 2862-2876. https://doi.org/10.1108/EC-03-2017-0090
Publisher
:Emerald Publishing Limited
Copyright © 2017, Emerald Publishing Limited