Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. 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. We first introduce and illustrate the Galerkin method as being applied to a system of linear equations formula_1 with the following symmetric  and positive definite matrix and the solution and right-hand-side vectors Let us take then the matrix of the Galerkin equation is the right-hand-side vector of the Galerkin equation is so that we obtain the solution vector to the Galerkin equation formula_8, which we finally uplift to determine the approximate solution to the original equation as In this example, our original Hilbert space is actually the 3-dimensional Euclidean space formula_10 equipped with the standard scalar product formula_11, our 3-by-3 matrix formula_12 defines the bilinear form formula_13, and the right-hand-side vector formula_14 defines the  bounded linear functional formula_15. The columns of the matrix formula_17 form an orthonormal basis of the 2-dimensional subspace of the Galerkin projection. The entries of the 2-by-2 Galerkin matrix formula_18 are formula_19, while the components of the right-hand-side vector formula_20 of the Galerkin equation are formula_21. Finally, the approximate solution formula_22 is obtained from the components of the solution vector formula_23 of the Galerkin equation and the basis as formula_24. Linear equation in a Hilbert space. Weak formulation of a linear equation. Let us introduce Galerkin's method with an abstract problem posed as a weak  formulation on a Hilbert space formula_17, namely, Here, formula_28 is a bilinear form (the exact requirements on formula_28 will be specified later) and formula_30 is a bounded linear functional on formula_17. Galerkin dimension reduction. Choose a subspace formula_32 of dimension "n" and solve the projected problem: We call this the Galerkin 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 compute formula_35 as a finite linear combination of the basis vectors in formula_36. Galerkin orthogonality. The key property of the Galerkin approach  is that the error is orthogonal to the chosen subspaces. Since formula_32, we can use formula_38 as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, formula_39 which is the error between the solution of the original problem, formula_40, and the solution of the Galerkin equation, formula_41 Matrix form of Galerkin's equation. 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 algorithmically. Let formula_43 be a basis for formula_44 . Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find formula_45 such that We expand formula_41 with respect to this basis, formula_48 and insert it into the equation above, to obtain This previous equation is actually a linear system of equations formula_50, where Symmetry 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 formula_28 is symmetric. Analysis of Galerkin methods. Here, we will restrict ourselves to symmetric bilinear forms, that is 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 of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution formula_41. The analysis will mostly rest on two properties of the bilinear form, namely By the Lax-Milgram theorem (see weak 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. Since formula_32, boundedness and ellipticity of the bilinear form apply to formula_44. 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 formula_63 between the original and the Galerkin solution admits the estimate This means, that up to the constant formula_65, the Galerkin solution formula_41 is as  close to the original solution formula_40 as any other vector in formula_44. In particular, it will be sufficient to study approximation by spaces formula_44, 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 formula_70: Dividing by formula_72 and taking the infimum over all possible formula_38 yields the lemma. Galerkin's best approximation property in the energy norm. For simplicity of presentation in  the section above we have assumed that the bilinear form formula_74 is symmetric and positive definite, which implies that it is a scalar product and the expression formula_75 is actually a valid vector norm, called the "energy norm". Under these assumptions one can easily prove in addition Galerkin's best approximation property in the energy norm. Using Galerkin a-orthogonality and the Cauchy–Schwarz inequality for the energy norm, we obtain Dividing by formula_77 and taking the infimum over all possible formula_70 proves that the Galerkin approximation formula_33 is the best approximation in the energy norm within the subspace formula_32 , i.e. formula_33 is nothing but the orthogonal, with respect to the scalar product formula_74, projection of the solution formula_40 to the subspace formula_44. Galerkin method for stepped Structures. I. Elishakof, M. Amato, A. Marzani, P.A. Arvan, and J.N. Reddy 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. The approach is usually credited to Boris Galerkin. The method was explained to the Western reader by Hencky and Duncan among others. Its convergence was studied  by Mikhlin and Leipholz Its coincidence with Fourier method was illustrated by Elishakoff et al. Its equivalence to Ritz's method for conservative problems was shown by Singer. Gander and Wanner showed how Ritz and Galerkin methods led to the modern finite element method. One hundred years of method's development was discussed by Repin. Elishakoff, Kaplunov and Kaplunov show that the Galerkin’s method was not developed by Ritz, contrary to the Timoshenko’s statements.