In mathematics, in the area of numerical analysis, Galerkin methods are a class of methods for converting a continuous operator problem (such as a differential equation) 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 but the method was discovered by Walther Ritz, to whom Galerkin refers. Often when referring to a Galerkin method, one also gives the name along with typical approximation methods used, such as Bubnov–Galerkin method (after Ivan Bubnov), Petrov–Galerkin method (after Georgii I. Petrov) or Ritz–Galerkin method (after Walther Ritz).
Examples of Galerkin methods are:
Introduction with an abstract problem
A problem in weak formulation
- find such that for all .
Here, is a bilinear form (the exact requirements on will be specified later) and is a bounded linear functional on .
Galerkin dimension reduction
Choose a subspace of dimension n and solve the projected problem:
- Find such that for all .
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
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 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 (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 , 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.
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.
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