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Collocation, Galerkin, and Rayleigh–Ritz Methods

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Boundary Value Problems for Engineers

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Abstract

Weighted residual methods (WRM) (also called Petrov-Galerkin methods ) provide simple and highly accurate solutions of BVPs. Collocation, Galerkin, and Rayleigh–Ritz methods are examples of the WRMs.

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Notes

  1. 1.

    Here, we do not deal with the other types of WRMs, such as Subdomain Method, Least-squares Method and Method of Moments.

  2. 2.

    The collocation method is used as an introduction of the concept of a residual , which leads to the Galerkin weighted residual method. This method is usually less accurate than the Galerkin or Rayleigh-Ritz Methods. If the governing ODE is known, then we apply the Galerkin (weighted residual ) approach, as in fluid mechanics and heat transfer problems. The Galerkin’s approach is usually more straightforward than the Rayleigh-Ritz approach.

  3. 3.

    Boris Grigorievich Galerkin (1871–1945) Russian Mathematician. Galerkin’s method was published in Russian in 1915 [6]. The detailed history of this method is found in [7, 8].

  4. 4.

    Walther Ritz (1878–1909) was a Swiss physicist who introduced this method [13] for problems involving elastic plates.

  5. 5.

    Note that the use of Rayleigh–Ritz Method is restricted to symmetric BVPs. The Galerkin Method is more general and does not require symmetry of the BVP.

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  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

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Keskin, A.Ü. (2019). Collocation, Galerkin, and Rayleigh–Ritz Methods. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_8

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  • DOI: https://doi.org/10.1007/978-3-030-21080-9_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21079-3

  • Online ISBN: 978-3-030-21080-9

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