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Galerkin method

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Inmathematics,in the area ofnumerical analysis,Galerkin methodsare a family of methods for converting a continuous operator problem, such as adifferential equation,commonly in aweak formulation,to a discrete problem by applying linear constraints determined by finite sets of basis functions. They are named after the Soviet mathematicianBoris Galerkin.

Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used:

Examples of Galerkin methods are:

Example: Matrix linear system

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We first introduce and illustrate the Galerkin method as being applied to a system of linear equations.We define the parameters as follow:

which is symmetric and positive definite, and the right-hand-side

The true solution to this linear system is

With Galerkin method, we can solve the system in a lower-dimensional space to obtain an approximate solution. Let us use the following basis for the subspace:

Then, we can write the Galerkin equationwhere the left-hand-side matrix is

and the right-hand-side vector is

We can then obtain the solution vector in the subspace:

which we finally project back to the original space to determine the approximate solution to the original equation as

In this example, our originalHilbert spaceis actually the 3-dimensional Euclidean spaceequipped with the standard scalar product,our 3-by-3 matrixdefines thebilinear form,and the right-hand-side vectordefines thebounded linear functional.The columns

of the matrixform an orthonormal basis of the 2-dimensional subspace of the Galerkin projection. The entries of the 2-by-2 Galerkin matrixare,while the components of the right-hand-side vectorof the Galerkin equation are.Finally, the approximate solutionis obtained from the components of the solution vectorof the Galerkin equation and the basis as.

Linear equation in a Hilbert space

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Weak formulation of a linear equation

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Let us introduce Galerkin's method with an abstract problem posed as aweak formulationon aHilbert space,namely,

findsuch that for all.

Here,is abilinear form(the exact requirements onwill be specified later) andis a bounded linear functional on.

Galerkin dimension reduction

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Choose a subspaceof dimensionnand solve the projected problem:

Findsuch that for all.

We call this theGalerkin equation.Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically computeas a finite linear combination of the basis vectors in.

Galerkin orthogonality

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The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since,we can useas a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error,which is the error between the solution of the original problem,,and the solution of the Galerkin equation,

Matrix form of Galerkin's equation

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Since the aim of Galerkin's method is the production of alinear system of equations,we build its matrix form, which can be used to compute the solution algorithmically.

Letbe abasisfor.Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: findsuch that

We expandwith respect to this basis,and insert it into the equation above, to obtain

This previous equation is actually a linear system of equations,where

Symmetry of the matrix

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Due to the definition of the matrix entries, the matrix of the Galerkin equation issymmetricif and only if the bilinear formis symmetric.

Analysis of Galerkin methods

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Here, we will restrict ourselves to symmetricbilinear forms,that is

While this is not really a restriction of Galerkin methods, the application of the standard theory becomes much simpler. Furthermore, aPetrov–Galerkin methodmay be required in the nonsymmetric case.

The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is awell-posed problemin the sense ofHadamardand therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution.

The analysis will mostly rest on two properties of thebilinear form,namely

  • Boundedness: for allholds
    for some constant
  • Ellipticity: for allholds
    for some constant

By the Lax-Milgram theorem (seeweak formulation), these two conditions imply well-posedness of the original problem in weak formulation. All norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm).

Well-posedness of the Galerkin equation

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Since,boundedness and ellipticity of the bilinear form apply to.Therefore, the well-posedness of the Galerkin problem is actually inherited from the well-posedness of the original problem.

Quasi-best approximation (Céa's lemma)

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The errorbetween the original and the Galerkin solution admits the estimate

This means, that up to the constant,the Galerkin solution is as close to the original solutionas any other vector in.In particular, it will be sufficient to study approximation by spaces,completely forgetting about the equation being solved.

Proof

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Since the proof is very simple and the basic principle behind all Galerkin methods, we include it here: by ellipticity and boundedness of the bilinear form (inequalities) and Galerkin orthogonality (equals sign in the middle), we have for arbitrary:

Dividing byand taking the infimum over all possibleyields the lemma.

Galerkin's best approximation property in the energy norm

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For simplicity of presentation in the section above we have assumed that the bilinear formis symmetric and positive definite, which implies that it is ascalar productand the expressionis actually a valid vector norm, called theenergy norm.Under these assumptions one can easily prove in addition Galerkin's best approximation property in the energy norm.

Using Galerkin a-orthogonality and theCauchy–Schwarz inequalityfor the energy norm, we obtain

Dividing byand taking the infimum over all possibleproves that the Galerkin approximationis the best approximation in the energy norm within the subspace,i.e.is nothing but the orthogonal, with respect to thescalar product,projection of the solutionto the subspace.

Galerkin method for stepped Structures

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I. Elishakof,M. Amato, A. Marzani, P.A. Arvan, and J.N. Reddy [6] [7] [8] [9] studied the application of the Galerkin method to stepped structures. They showed that the generalized function, namely unit-step function, Dirac’s delta function, and the doublet function are needed for obtaining accurate results.

History

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The approach is usually credited toBoris Galerkin.[10][11]The method was explained to the Western reader by Hencky[12]and Duncan[13][14]among others. Its convergence was studied by Mikhlin[15]and Leipholz[16][17][18][19]Its coincidence with Fourier method was illustrated byElishakoffet al.[20][21][22]Its equivalence to Ritz's method for conservative problems was shown by Singer.[23]Gander and Wanner[24]showed how Ritz and Galerkin methods led to the modern finite element method. One hundred years of method's development was discussed by Repin.[25]Elishakoff, Kaplunov and Kaplunov[26]show that the Galerkin’s method was not developed by Ritz, contrary to the Timoshenko’s statements.

See also

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References

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  1. ^A. Ern, J.L. Guermond,Theory and practice of finite elements,Springer, 2004,ISBN0-387-20574-8
  2. ^"Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
  3. ^S. Brenner, R. L. Scott,The Mathematical Theory of Finite Element Methods,2nd edition, Springer, 2005,ISBN0-387-95451-1
  4. ^P. G. Ciarlet,The Finite Element Method for Elliptic Problems,North-Holland, 1978,ISBN0-444-85028-7
  5. ^Y. Saad,Iterative Methods for Sparse Linear Systems,2nd edition, SIAM, 2003,ISBN0-89871-534-2
  6. ^Elishakoff, I., Amato, M., Ankitha, A. P., & Marzani, A. (2021). Rigorous implementation of the Galerkin method for stepped structures needs generalized functions. Journal of Sound and Vibration, 490, 115708.
  7. ^Elishakoff, I., Amato, M., & Marzani, A. (2021). Galerkin’s method revisited and corrected in the problem of Jaworsky and Dowell. Mechanical Systems and Signal Processing, 155, 107604.
  8. ^Elishakoff, I., & Amato, M. (2021). Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient. International Journal of Mechanics and Materials in Design, 1-17.
  9. ^Amato, M., Elishakoff, I., & Reddy, J. N. (2021). Flutter of a Multicomponent Beam in a Supersonic Flow. AIAA Journal, 59(11), 4342-4353.
  10. ^Galerkin, B.G.,1915, Rods and Plates, Series Occurring in Various Questions Concerning the Elastic Equilibrium of Rods and Plates, Vestnik Inzhenerov i Tekhnikov, (Engineers and Technologists Bulletin), Vol. 19, 897-908 (in Russian),(English Translation: 63-18925, Clearinghouse Fed. Sci. Tech. Info.1963).
  11. ^"Le destin douloureux de Walther Ritz (1878-1909)", (Jean-Claude Pont, editor), Cahiers de Vallesia, 24, (2012),ISBN978-2-9700636-5-0
  12. ^Hencky H.,1927, Eine wichtige Vereinfachung der Methode von Ritz zur angennäherten Behandlung von Variationproblemen, ZAMM: Zeitschrift für angewandte Mathematik und Mechanik, Vol. 7, 80-81 (in German).
  13. ^Duncan, W.J.,1937, Galerkin’s Method in Mechanics and Differential Equations, Aeronautical Research Committee Reports and Memoranda, No. 1798.
  14. ^Duncan, W.J., 1938, The Principles of the Galerkin Method, Aeronautical Research Report and Memoranda, No. 1894.
  15. ^S. G. Mikhlin, "Variational methods in Mathematical Physics", Pergamon Press, 1964
  16. ^Leipholz H.H.E., 1976, Use of Galerkin’s Method for Vibration Problems, Shock and Vibration Digest, Vol. 8, 3-18
  17. ^Leipholz H.H.E., 1967, Über die Wahl der Ansatzfunktionen bei der Durchführung des Verfahrens von Galerkin, Acta Mech., Vol. 3, 295-317 (in German).
  18. ^Leipholz H.H.E., 1967, Über die Befreiung der Anzatzfunktionen des Ritzschen und Galerkinschen Verfahrens von den Randbedingungen, Ing. Arch., Vol. 36, 251-261 (in German).
  19. ^Leipholz, H.H.E.,1976, Use of Galerkin’s Method for Vibration Problems, The Shock and Vibration Digest Vol. 8, 3-18, 1976.
  20. ^Elishakoff, I., Lee, L.H.N.,1986, On Equivalence of the Galerkin and Fourier Series Methods for One Class of Problems, Journal of Sound and Vibration, Vol. 109, 174-177.
  21. ^Elishakoff, I., Zingales, M., 2003, Coincidence of Bubnov-Galerkin and Exact Solution in an Applied Mechanics Problem, Journal of Applied Mechanics, Vol. 70, 777-779.
  22. ^Elishakoff, I., Zingales M., 2004, Convergence of Bubnov-Galerkin Method Exemplified, AIAA Journal, Vol. 42(9), 1931-1933.
  23. ^Singer J., 1962, On Equivalence of the Galerkin and Rayleigh-Ritz Methods, Journal of the Royal Aeronautical Society, Vol. 66, No. 621, p.592.
  24. ^Gander, M.J, Wanner, G., 2012, From Euler, Ritz, and Galerkin to Modern Computing, SIAM Review, Vol. 54(4), 627-666.
  25. ^] Repin, S., 2017, One Hundred Years of the Galerkin Method, Computational Methods and Applied Mathematics, Vol. 17(3), 351-357.
  26. ^.Elishakoff, I., Julius Kaplunov, Elizabeth Kaplunov, 2020, “Galerkin’s method was not developed by Ritz, contrary to the Timoshenko’s statement”, in Nonlinear Dynamics of Discrete and Continuous Systems (A. Abramyan, I. Andrianov and V. Gaiko, eds.), pp. 63-82, Springer, Berlin.
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