research-article

Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz Problems

Authors:

Bernard Bialecki,

Graeme Fairweather,

Andreas KarageorghisAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 24, Issue 5

Pages 1733 - 1753

https://doi.org/10.1137/S106482750139964X

Published: 01 January 2003 Publication History

Abstract

We consider the solution of various boundary value problems for the Helmholtz equation in the unit square using a nodal cubic spline collocation method and modifications of it which produce optimal (fourth-) order approximations. For the solution of the collocation equations, we formulate matrix decomposition algorithms, fast direct methods which employ fast Fourier transforms and require O(N² log N) operations on an N × N uniform partition of the unit square. A computational study confirms the published analysis for the Dirichlet problem and indicates that similar results hold for Neumann, mixed, and periodic boundary conditions. The numerical results also exhibit superconvergence phenomena not reported in earlier studies.

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Cited By

Du KFairweather GNguyen QSun W(2022)Matrix decomposition algorithms for the C 0-quadratic finite element Galerkin methodBIT10.1007/s10543-009-0233-049:3(509-526)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-009-0233-0
Bialecki BFairweather GKarageorghis ANguyen Q(2022)Optimal superconvergent one step quadratic spline collocation methods BIT10.1007/s10543-008-0188-648:3(449-472)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-008-0188-6
Bialecki BFairweather GKarageorghis AMaack J(2019)A quadratic spline collocation method for the Dirichlet biharmonic problemNumerical Algorithms10.1007/s11075-019-00676-z83:1(165-199)Online publication date: 19-Feb-2019
https://dl.acm.org/doi/10.1007/s11075-019-00676-z
Show More Cited By

Index Terms

Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz Problems
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 24, Issue 5

2003

373 pages

ISSN:1064-8275

Issue’s Table of Contents

Copyright © 2003 Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2003

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Cited By

Du KFairweather GNguyen QSun W(2022)Matrix decomposition algorithms for the C 0-quadratic finite element Galerkin methodBIT10.1007/s10543-009-0233-049:3(509-526)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-009-0233-0
Bialecki BFairweather GKarageorghis ANguyen Q(2022)Optimal superconvergent one step quadratic spline collocation methods BIT10.1007/s10543-008-0188-648:3(449-472)Online publication date: 11-Mar-2022
https://dl.acm.org/doi/10.1007/s10543-008-0188-6
Bialecki BFairweather GKarageorghis AMaack J(2019)A quadratic spline collocation method for the Dirichlet biharmonic problemNumerical Algorithms10.1007/s11075-019-00676-z83:1(165-199)Online publication date: 19-Feb-2019
https://dl.acm.org/doi/10.1007/s11075-019-00676-z
Bialecki BWright L(2015)A fast direct solver for a fourth order finite difference scheme for Poisson's equation on the unit disc in polar coordinatesNumerical Algorithms10.1007/s11075-015-9971-z70:4(727-751)Online publication date: 1-Dec-2015
https://dl.acm.org/doi/10.1007/s11075-015-9971-z
Bialecki BKarageorghis A(2013)Modified nodal cubic spline collocation for three-dimensional variable coefficient second order partial differential equationsNumerical Algorithms10.1007/s11075-012-9669-464:2(349-383)Online publication date: 1-Oct-2013
https://dl.acm.org/doi/10.1007/s11075-012-9669-4
Bialecki BFairweather GKarageorghis A(2011)Matrix decomposition algorithms for elliptic boundary value problemsNumerical Algorithms10.1007/s11075-010-9384-y56:2(253-295)Online publication date: 1-Feb-2011
https://dl.acm.org/doi/10.1007/s11075-010-9384-y
Bialecki BFairweather GKarageorghis A(2005)Optimal Superconvergent One Step Nodal Cubic Spline Collocation MethodsSIAM Journal on Scientific Computing10.1137/04060979327:2(575-598)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1137/040609793

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