Galerkin method

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 to a function space, by converting the equation to a weak formulation. Typically one then applies some constraints on the functions space to characterize the space with a finite set of basis functions. Often when using a Galerkin method one also gives the name along with typical approximation methods used, such as Petrov-Galerkin method or Ritz-Galerkin method.A. Ern, J.L. Guermond, "Theory and practice of finite elements", Springer, 2004, ISBN 0-3872-0574-8 ]

The approach is credited to the Russian mathematician Boris Galerkin.

Since the beauty of Galerkin methods lies in the very abstract way of studying them, we will first give their abstract derivation. In the end, we will give examples for their use.

Examples for Galerkin methods are:
* The finite element methodS. Brenner, R. L. Scott, "The Mathematical Theory of Finite Element Methods", 2nd edition, Springer, 2005, ISBN 0-3879-5451-1 ] , P. G. Ciarlet, "The Finite Element Method for Elliptic Problems", North-Holland, 1978, ISBN 0-4448-5028-7 ]
* Boundary element method for solving integral equations
* Krylov subspace methods Y. Saad, "Iterative Methods for Sparse Linear Systems", 2nd edition, SIAM, 2003, ISBN 0-8987-1534-2 ]

Introduction with an abstract problem

A problem in weak formulation

Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space, $V$, namely,: find $uin\; V$ such that for all $vin\; V$, $a(u,v)\; =\; f(v)$.

Here, $a(cdot,cdot)$ is a bilinear form (the exact requirements on $a(cdot,cdot)$ will be specified later) and $f$ is a bounded linear operator on $V$.

Galerkin discretization

Choose a subspace $V\_n\; subset\; V$, dimension "n" and solve the projected problem:: Find $u\_nin\; V\_n$ such that for all $v\_nin\; V\_n$, $a(u\_n,v\_n)\; =\; f(v\_n)$.

We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed.

Galerkin orthogonality

The key property of the Galerkin approach is that the error is orthogonal to the chosen subspace. Since $V\_n\; subset\; V$, we can use $v\_n$ as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, $e\_n\; =\; u-u\_n$ which is the error between the solution of the original problem, $u$, and the solution of the Galerkin equation, $u\_n$

:$a(e\_n,\; v\_n)\; =\; a(u,v\_n)\; -\; a(u\_n,\; v\_n)\; =\; f(v\_n)\; -\; f(v\_n)\; =\; 0.$

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 by a computer program.

Let $e\_1,\; e\_2,ldots,e\_n$ be a basis for $V\_n$. Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find $u\_n\; in\; V\_n$ such that

:$a(u\_n,\; e\_i)\; =\; f(e\_i)\; quad\; i=1,ldots,n.$

We expand $u\_n$ in respect to this basis, $u\_n\; =\; sum\_\{j=1\}^n\; u\_je\_j$ and insert it into the equation above, to obtain

:$aleft(sum\_\{j=1\}^n\; u\_je\_j,\; e\_i\; ight)\; =\; sum\_\{j=1\}^n\; u\_j\; a(e\_j,\; e\_i)\; =\; f(e\_i)\; quad\; i=1,ldots,n.$

This previous equation is actually a linear system of equations $Au=f$, where

:$a\_\{ij\}\; =\; a(e\_j,\; e\_i),\; quad\; f\_i\; =\; f(e\_i).$

ymmetry 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 $a(cdot,cdot)$ is symmetric.

Analysis of Galerkin methods

Here, we will restrict ourselves to symmetric bilinear forms, that is

:$a(u,v)\; =\; a(v,u).$

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 $u\_n$.

The analysis will mostly rest on two properties of the bilinear form, namely
* Boundedness: for all $u,vin\; V$ holds
*:$a(u,v)\; le\; C\; |u|,\; |v|$ for some constant $C>0$
* Ellipticity: for all $uin\; V$ holds
*:$a(u,u)\; ge\; c\; |u|^2$ for some constant $c>0$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 energy norm).

Well-posedness of the Galerkin equation

Since $V\_n\; subset\; V$, boundedness and ellipticity of the bilinear form apply to $V\_n$. 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 $e\_n\; =\; u-u\_n$ between the original and the Galerkin solution admits the estimate

:$|e\_n|\; le\; frac\{C\}\{c\}\; inf\_\{v\_nin\; V\_n\}\; |u-v\_n|.$

This means, that up to the constant $C/c$, the Galerkin solution $u\_n$is as close to the original solution $u$ as any other vector in $V\_n$. In particular, it will be sufficient to study approximation by spaces $V\_n$, 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 $v\_nin\; V\_n$:

:$c|e\_n|^2\; le\; a(e\_n,\; e\_n)\; =\; a(e\_n,\; u-v\_n)\; le\; C\; |e\_n|\; ,\; |u-v\_n|.$

Dividing by $c\; |e\_n|$ and taking the infimum over all possible $v\_h$ yields the lemma.

Application to the finite element method for Poisson's equation

Application to the analysis of the conjugate gradient method

References

External links

* [http://math.fullerton.edu/mathews/n2003/GalerkinMod.html Galerkin's Method]
* [http://mathworld.wolfram.com/GalerkinMethod.html Galerkin Method from MathWorld]