dc.contributor.authorZhang, Jing
dc.contributor.authorWang, Li-Lian
dc.contributor.authorLi, Huiyuan
dc.contributor.authorZhang, Zhimin
dc.identifier.citationZhang, J., Wang, L.-L., Li, H., & Zhang, Z. (2017). Optimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order -1. Journal of Scientific Computing, 70(2), 451-477.en_US
dc.description.abstractWe introduce a family of generalized prolate spheroidal wave functions (PSWFs) of order -1, and develop new spectral schemes for second-order boundary value problems. Our technique differs from the differentiation approach based on PSWFs of order zero in Kong and Rokhlin (Appl Comput Harmon Anal 33(2):226–260, 2012); in particular, our orthogonal basis can naturally include homogeneous boundary conditions without the re-orthogonalization of Kong and Rokhlin (2012). More notably, it leads to diagonal systems or direct “explicit” solutions to 1D Helmholtz problems in various situations. Using a rule optimally pairing the bandwidth parameter and the number of basis functions as in Kong and Rokhlin (2012), we demonstrate that the new method significantly outperforms the Legendre spectral method in approximating highly oscillatory solutions. We also conduct a rigorous error analysis of this new scheme. The idea and analysis can be extended to generalized PSWFs of negative integer order for higher-order boundary value and eigenvalue problems.en_US
dc.description.sponsorshipMOE (Min. of Education, S’pore)en_US
dc.relation.ispartofseriesJournal of Scientific Computingen_US
dc.rights© 2017 Springer.en_US
dc.subjectGeneralized prolate spheroidal wave functions of order -1en_US
dc.subjectHelmholtz Equationsen_US
dc.titleOptimal Spectral Schemes Based on Generalized Prolate Spheroidal Wave Functions of Order -1en_US
dc.typeJournal Article
dc.contributor.schoolSchool of Physical and Mathematical Sciencesen_US

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