In mathematics, in the area of numerical analysis, Galerkin methods, named for 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:
Here, is a bilinear form (the exact requirements on will be specified later) and is a bounded linear functional on .
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 .
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,
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 symmetric if and only if the bilinear form is symmetric.
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
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).
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.
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.
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 by and taking the infimum over all possible yields the lemma.
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.
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.