- Galerkin method
In
mathematics , in the area ofnumerical analysis , Galerkin methods are a class of methods for converting a continuous operator problem (such as adifferential equation ) to a discrete problem. In principle, it is the equivalent of applying themethod of variation to a function space, by converting the equation to aweak formulation . Typically one then applies some constraints on the functions space to characterize the space with a finite set of basis functions. Often when using a Galerkin method one also gives the name along with typical approximation methods used, such asPetrov-Galerkin method orRitz-Galerkin method .A. Ern, J.L. Guermond, "Theory and practice of finite elements", Springer, 2004, ISBN 0-3872-0574-8 ]The approach is credited to the Russian mathematician
Boris Galerkin .Since the beauty of Galerkin methods lies in the very abstract way of studying them, we will first give their abstract derivation. In the end, we will give examples for their use.
Examples for Galerkin methods are:
* Thefinite element method S. Brenner, R. L. Scott, "The Mathematical Theory of Finite Element Methods", 2nd edition, Springer, 2005, ISBN 0-3879-5451-1 ] , P. G. Ciarlet, "The Finite Element Method for Elliptic Problems", North-Holland, 1978, ISBN 0-4448-5028-7 ]
*Boundary element method for solving integral equations
*Krylov subspace method s Y. Saad, "Iterative Methods for Sparse Linear Systems", 2nd edition, SIAM, 2003, ISBN 0-8987-1534-2 ]Introduction with an abstract problem
A problem in weak formulation
Let us introduce Galerkin's method with an abstract problem posed as a
weak formulation on aHilbert space , V, namely,: find uin V such that for all vin V, a(u,v) = f(v).Here, a(cdot,cdot) is a
bilinear form (the exact requirements on a(cdot,cdot) will be specified later) and f is abounded linear operator on V.Galerkin discretization
Choose a subspace V_n subset V, dimension "n" and solve the projected problem:: Find u_nin V_n such that for all v_nin V_n, a(u_n,v_n) = f(v_n).
We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed.
Galerkin orthogonality
The key property of the Galerkin approach is that the error is orthogonal to the chosen subspace. Since V_n subset V, we can use v_n as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, e_n = u-u_n which is the error between the solution of the original problem, u, and the solution of the Galerkin equation, u_n
:a(e_n, v_n) = a(u,v_n) - a(u_n, v_n) = f(v_n) - f(v_n) = 0.
Matrix form
Since the aim of Galerkin's method is the production of a linear system of equations, we build its matrix form, which can be used to compute the solution by a computer program.
Let e_1, e_2,ldots,e_n be a basis for V_n. Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find u_n in V_n such that
:a(u_n, e_i) = f(e_i) quad i=1,ldots,n.
We expand u_n in respect to this basis, u_n = sum_{j=1}^n u_je_j and insert it into the equation above, to obtain
:aleft(sum_{j=1}^n u_je_j, e_i ight) = sum_{j=1}^n u_j a(e_j, e_i) = f(e_i) quad i=1,ldots,n.
This previous equation is actually a linear system of equations Au=f, where
:a_{ij} = a(e_j, e_i), quad f_i = f(e_i).
ymmetry of the matrix
Due to the definition of the matrix entries, the matrix of the Galerkin equation is symmetric if and only if the bilinear form a(cdot,cdot) is symmetric.
Analysis of Galerkin methods
Here, we will restrict ourselves to symmetric
bilinear form s, that is:a(u,v) = a(v,u).
While this is not really a restriction of Galerkin methods, the application of the standard theory becomes much simpler. Furthermore, a
Petrov-Galerkin method may be required in the nonsymmetric case.The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a
well-posed problem in the sense ofHadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution u_n.The analysis will mostly rest on two properties of the
bilinear form , namely
* Boundedness: for all u,vin V holds
*:a(u,v) le C |u|, |v| for some constant C>0
* Ellipticity: for all uin V holds
*:a(u,u) ge c |u|^2 for some constant c>0By 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 energy norm).Well-posedness of the Galerkin equation
Since V_n subset V, boundedness and ellipticity of the bilinear form apply to V_n. 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)
The error e_n = u-u_n between the original and the Galerkin solution admits the estimate
:e_n| le frac{C}{c} inf_{v_nin V_n} |u-v_n|.
This means, that up to the constant C/c, the Galerkin solution u_nis as close to the original solution u as any other vector in V_n. In particular, it will be sufficient to study approximation by spaces V_n, completely forgetting about the equation being solved.
Proof
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 v_nin V_n:
:c|e_n|^2 le a(e_n, e_n) = a(e_n, u-v_n) le C |e_n| , |u-v_n|.
Dividing by c |e_n| and taking the infimum over all possible v_h yields the lemma.
Application to the finite element method for Poisson's equation
Application to the analysis of the conjugate gradient method
References
External links
* [http://math.fullerton.edu/mathews/n2003/GalerkinMod.html Galerkin's Method]
* [http://mathworld.wolfram.com/GalerkinMethod.html Galerkin Method from MathWorld]
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