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 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 the Russian mathematician Boris Galerkin but the method was discovered by the Swiss mathematician Walther Ritz,[1] 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[2][3]) or Ritz–Galerkin method[4] (after Walther Ritz).

Examples of Galerkin methods are:

Introduction with an abstract problem[edit]

A problem in weak formulation[edit]

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).

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 functional on V.

Galerkin Dimension Reduction[edit]

Choose a subspace V_n \subset V of dimension n 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 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  u_n as a finite linear combination of the basis vectors in  V_n .

Galerkin orthogonality[edit]

The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. 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, \epsilon_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(\epsilon_n, v_n) = a(u,v_n) - a(u_n, v_n) = f(v_n) - f(v_n) = 0.

Matrix form[edit]

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 with 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).

Symmetry of the matrix[edit]

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[edit]

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\| 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 an energy norm).

Well-posedness of the Galerkin equation[edit]

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)[edit]

Main article: Céa's lemma

The error u-u_n between the original and the Galerkin solution admits the estimate

\|u-u_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_n. In particular, it will be sufficient to study approximation by spaces V_n, completely forgetting about the equation being solved.

Proof[edit]

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\|u-u_n\|^2 \le a(u-u_n, u-u_n) = a(u-u_n, u-v_n) \le C \|u-u_n\| \, \|u-v_n\|.

Dividing by c \|u-u_n\| and taking the infimum over all possible v_n yields the lemma.

References[edit]

  1. ^ "Le destin douloureux de Walther Ritz (1878-1909)", (Jean-Claude Pont, editor), Cahiers de Vallesia, 24, (2012), ISBN 978-2-9700636-5-0
  2. ^ S. G. Mikhlin, "Variational methods in Mathematical Physics", Pergamon Press, 1964
  3. ^ "Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
  4. ^ A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0-387-20574-8
  5. ^ S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0-387-95451-1
  6. ^ P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978, ISBN 0-444-85028-7
  7. ^ Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0-89871-534-2

External links[edit]