Abstract
We develop an H m-conforming (m ≥ 1) spectral element method on multi-dimensional domain associated with the partition into multi-dimensional rectangles. We construct a set of basis functions on the interval [−1, 1] that are made up of the generalized Jacobi polynomials (GJPs) and the nodal basis functions. So the basis functions on multi-dimensional rectangles consist of the tensorial product of the basis functions on the interval [−1, 1]. Then we construct the spectral element interpolation operator and prove the associated interpolation error estimates. Finally, we apply the H 2-conforming spectral element method to the Helmholtz transmission eigenvalues that is a hot problem in the field of engineering and mathematics.
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Acknowledgements
This work was supported by the Educational Innovation Program of Guizhou Province for Graduate Students (Grant No. KYJJ[2016]01) and National Natural Science Foundation of China (Grant No. 11561014). The authors cordially thank the referees for their valuable comments and suggestions that led to the large improvement of this paper.
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Han, J., Yang, Y. An H m-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues. Sci. China Math. 60, 1529–1542 (2017). https://doi.org/10.1007/s11425-015-0847-y
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DOI: https://doi.org/10.1007/s11425-015-0847-y
Keywords
- spectral element method
- multi-dimensional domain
- interpolation error estimates
- transmission eigenvalues