In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem 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 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 illustred 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 developement was discussed by Repin. Often when referring to a Galerkin method, one also gives the name along with typical approximation methods used, such as Bubnov–Galerkin method, Petrov–Galerkin method or Ritz–Galerkin method. Examples of Galerkin methods are:
Choose a subspace 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 as a finite linear combination of the basis vectors in.
Galerkin orthogonality
The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since , we can use as 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
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 be a basis for. Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find such that We expand with respect to this basis, and insert it into the equation above, to obtain This previous equation is actually a linear system of equations, where
Due to the definition of the matrix entries, the matrix of the Galerkin equation is symmetricif and only if the bilinear form 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. The analysis will mostly rest on two properties of the bilinear form, namely
Boundedness: for all holds
: for some constant
Ellipticity: for all holds
: for some constant
By the Lax-Milgram theorem, 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.
Well-posedness of the Galerkin equation
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)
The error between 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 solution as any other vector in. In particular, it will be sufficient to study approximation by spaces, 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 and Galerkin orthogonality, we have for arbitrary : Dividing by and taking the infimum over all possible yields the lemma.
