Galerkin method

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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.[1]

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:

Contents

[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 (the exact requirements on a(\cdot,\cdot) will be specified later) and f is a bounded linear operator 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(un,vn) = f(vn).

We will call this the Galerkin equation. Notice 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 vn as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error

a(en,vn) = a(u,vn) − a(un,vn) = f(vn) − f(vn) = 0.

Here, en = uun is the error between the solution of the original problem u and the Galerkin equation un, 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 Vn. 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 un in respect to this basis, u_n = \sum_{j=1}^n u_je_j and insert it into the equation above, to obtain

a\left(\sum_{j=1}^n u_je_j, e_i\right) = \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).

[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 un.

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\| for some constant C > 0
  • Ellipticity: for all u\in 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).

[edit] Well-posedness of the Galerkin equation

Since V_n \subset V, boundedness and ellipticity of the bilinear form apply to Vn. 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)

Main article: Céa's lemma

The error en = uun 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 un is as close to the original solution u as any other vector in Vn. In particular, it will be sufficient to study approximation by spaces Vn, 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 vh yields the lemma.

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


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


[edit] References

  1. ^ A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0-3872-0574-8
  2. ^ S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0-3879-5451-1
  3. ^ P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978, ISBN 0-4448-5028-7
  4. ^ Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0-8987-1534-2

[edit] External links