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| Content Provider | ACM Digital Library |
|---|---|
| Author | Boyd, John P. |
| Copyright Year | 2005 |
| Abstract | High order domain decomposition methods using a basis of Legendre polynomials, known variously as “spectral elements” or $“\textit{p}-type$ finite elements,” have become very popular. Recent studies suggest that accuracy and efficiency can be improved by replacing Legendre polynomials by prolate spheroidal wave functions of zeroth order. In this article, we explain the practicalities of computing all the numbers needed to switch bases: the grid points $x_{j},$ the quadrature weights $w_{j},$ and the values of the prolate functions and their derivatives at the grid points. The prolate functions themselves are computed by a Legendre-Galerkin discretization of the prolate differential equation; this yields a symmetric tridiagonal matrix. The prolate functions are then defined by Legendre series whose coefficients are the eigenfunctions of the matrix eigenproblem. The grid points and weights are found simultaneously through a Newton iteration. For large $\textit{N}$ and $\textit{c},$ the iteration diverges from a first guess of the Legendre-Lobatto points and weights. Fortunately, the variations of the $x_{j}$ and $w_{j}$ with $\textit{c}$ are well-approximated by a symmetric parabola over the whole range of interest. This makes it possible to bypass the continuation procedures of earlier authors. |
| Starting Page | 149 |
| Ending Page | 165 |
| Page Count | 17 |
| File Format | |
| ISSN | 00983500 |
| e-ISSN | 15577295 |
| DOI | 10.1145/1055531.1055538 |
| Volume Number | 31 |
| Issue Number | 1 |
| Journal | ACM Transactions on Mathematical Software (TOMS) |
| Language | English |
| Publisher | Association for Computing Machinery (ACM) |
| Publisher Date | 2005-03-01 |
| Publisher Place | New York |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | $\textit{p}-finite$ elements Prolate spheroidal wavefunctions Prolate element method Spectral elements |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics Software |
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