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Galerkin Methods

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Error Estimates for Advanced Galerkin Methods

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 88))

Abstract

Having derived in the previous chapter various boundary value problems, including the finite and linearized hyperelasticity problems for both compressible and (nearly) incompressible materials, a reasonable question is how these problems can be solved. For most cases in engineering practice, the problems, including their geometry, are too complex for the feasible derivation of an exact analytical solution even though such a solution exists. We are therefore forced to employ numerical methods to obtain, at least, approximate solutions to the boundary value problems stated in the previous chapter.

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Notes

  1. 1.

    In the literature, the Galerkin method is also referred to as the Rayleigh-Ritz-Galerkin or the Ritz-Galerkin method to honor the works of John William Strutt, 3rd Baron Rayleigh (1842–1919) and Walther Ritz (1878–1909) that serve as a basis for the Galerkin method.

  2. 2.

    The reader is reminded that both \({\mathcal T}\) and \({\mathcal V}\) were already introduced in Sect. 3.1.3.

  3. 3.

    The error measure \(E_2\) is an artificial error measure. Therefore, the factor 1/2 is optional and does not affect the value of \(\varvec{a}\). We include this factor because it provides consistency with quadratic functionals of a physical nature, such as virtually all energy functionals.

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    Marcus Olavi Rüter

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Rüter, M.O. (2019). Galerkin Methods. In: Error Estimates for Advanced Galerkin Methods. Lecture Notes in Applied and Computational Mechanics, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-06173-9_4

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