In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting 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.
They are named after the Soviet mathematician Boris Galerkin.
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 equation
where 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 original Hilbert space is actually the 3-dimensional Euclidean space
form an orthonormal basis of the 2-dimensional subspace of the Galerkin projection.
Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space
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.
Subtracting the two, we get the Galerkin orthogonality relation for the error,
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.
Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find
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.
, boundedness and ellipticity of the bilinear form apply to
between the original and the Galerkin solution admits the estimate This means, that up to the constant
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
For simplicity of presentation in the section above we have assumed that the bilinear form
is symmetric and positive definite, which implies that it is a scalar product and the expression
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
Reddy [6] [7] [8] [9] studied the application of the Galerkin method to stepped structures.
[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 by Elishakoff et 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.