Galerkin method

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In mathematics, in the area of numerical analysis, the Galerkin method is a means for converting a differential equation to a problem of linear algebra or a high dimensional linear system of equations, which may then be projected to a lower dimensional system. It relies on the weak formulation of an equation and works in principle by restricting the possible solutions as well as the test functions to a smaller space than the original one (see below for more details). These small systems are easier to solve than the original problem, but their solution is only an approximation to the original solution.

The approach was invented by 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 method
Method of moments for solving integral equations
Krylov subspace methods

1 Introduction with an abstract problem
2 Analysis of Galerkin methods
- 2.1 Well-posedness of the Galerkin equation
- 2.2 Quasi-best approximation (Céa's lemma)
  - 2.2.1 Proof
3 Application to the finite element method for Poisson's equation
4 Application to the analysis of the conjugate gradient method
5 Literature
6 External links

[edit] Introduction with an abstract problem

[edit] 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 $u\in V$ such that for all $v\in V$

a (u, v) = f (v)

holds. Here, $a(\cdot,\cdot)$ is a bilinear form and $f$ is a linear form on $V$ .

[edit] Galerkin discretization

Choose a subspace $V_n \subset V$ , which is of much smaller dimension (actually, we will assume that the index $n$ denotes its dimension) and solve the projected problem: find $u_n\in V_n$ such that for all $v_n\in V_n$

a (u n, v n) = f (v n).

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

[edit] Galerkin orthogonality

This is the key property making the mathematical analysis of Galerkin methods very sharp. 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

a (e n, v n) = a (u, v n) - a (u n, v n) = f (v n) - f (v n) = 0.

Here, $e n = u - u n$ is the error between the solution of the original problem $u$ and the Galerkin equation $u h$ , respectively.

[edit] 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 develop $u n$ with respect to this basis getting $u_n = \sum_{j=1}^n u_je_j$ and inserting it into the equation above

$a(\sum u_je_j, e_i) = \sum u_j a(e_j, e_i) = f(e_i) \quad i=1,\ldots,n.$

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

$a_{ij} = a(e_j, e_i), \quad f_i = f(e_i).$

[edit] 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 $a(\cdot,\cdot)$ is symmetric.

[edit] 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 $u,v\in V$ holds

$a(u,v) \le C \|u\|\, \|v\|$
Ellipticity: for all $u\in V$ holds

$a(u,u) \ge c \|u\|^2$

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 holds (these norms are often called energy norm).

[edit] 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.

[edit] Quasi-best approximation (Céa's lemma)

The error $e n = u - u n$ between the original and the Galerkin solution admits the estimate

$\|e_n\| \le \frac{C}{c} \inf_{v_n\in V_n} \|u-v_n\|.$

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 $V h$ . In particular, it will be sufficient to study approximation by spaces $V n$ , completely forgetting about the equation being solved.

[edit] 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 $v_n\in V_n$ :

$c\|e_n\|^2 \le a(e_n, e_n) = a(e_n, u-v_n) \le C \|e_n\| \, \|u-v_n\|.$

Dividing by $c \|e_n\|$ and taking the infimum over all possible $v h$ yields the lemma.

[edit] Application to the finite element method for Poisson's equation

still to come

[edit] Application to the analysis of the conjugate gradient method

still to come

[edit] Literature

Usually, Galerkin methods are not a topic alone in literature. They are discussed alongside their applications. Therefore, the reader is referred to textbooks on the finite element method.

Here, we would like to name

P. G. Ciarlet: The Finite Element Method for Elliptic Problems, North-Holland, 1978

The analysis of Krylov space methods in this framework can be found in

Y. Saad: Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003

[edit] External links

Retrieved from "http://en.wikipedia.org../../../g/a/l/Galerkin_method.html"

Categories: Cleanup from November 2006 | All pages needing cleanup | Numerical differential equations