Petrov-Galerkin method

In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) to a discrete problem. In principle, it is the equivalent of applying the method of variation of parameters to a function space, by converting the equation to a weak formulation. Typically one then applies some constraints on the function space to characterize the space with a finite set of basis functions. The approach is usually credited to the Russian mathematician Boris Galerkin but the method was discovered by the Swiss mathematician Walther Ritz,[1] to whom Galerkin refers. Often when referring to a Galerkin method, one also gives the name along with typical approximation methods used, such as Bubnov–Galerkin method (after Ivan Bubnov), Petrov–Galerkin method (after Georgii I. Petrov [2][3]) or Ritz–Galerkin method [4] (after Walther Ritz).

Examples of Galerkin methods are:
  • - the Galerkin method of weighted residuals, the most common method of calculating the global stiffness matrix in the finite element method,[5][6]
  • - the boundary element method for solving integral equations,
  • - Krylov subspace methods.[7]
  • [1] "Le destin douloureux de Walther Ritz (1878-1909)", (Jean-Claude Pont, editor), Cahiers de Vallesia, 24, (2012), ISBN 978-2-9700636-5-0
  • [2] S. G. Mikhlin, "Variational methods in Mathematical Physics", Pergamon Press, 1964
  • [3] "Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
  • [4] A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0-387-20574-8
  • [5] S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0-387-95451-1
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  • [6] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978, ISBN 0-444-85028-7
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  • [7] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0-89871-534-2