In

Hazewinkel, Michiel, ed. (2001), "Galerkin method",

Galerkin's Method

Galerkin Method from MathWorld
External links

^ "Le destin douloureux de Walther Ritz (18781909)", (JeanClaude Pont, editor), Cahiers de Vallesia, 24, (2012), ISBN 9782970063650

^ S. G. Mikhlin, "Variational methods in Mathematical Physics", Pergamon Press, 1964

^ "Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289291, DOI 10.1134/S0015462812030015

^ A. Ern, J.L. Guermond, Theory and practice of finite elements, Springer, 2004, ISBN 0387205748

^ S. Brenner, R. L. Scott, The Mathematical Theory of Finite Element Methods, 2nd edition, Springer, 2005, ISBN 0387954511

^ P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, 1978, ISBN 0444850287

^ Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, 2003, ISBN 0898715342
References
Dividing by c \uu_n\ and taking the infimum over all possible v_n yields the lemma.

c\uu_n\^2 \le a(uu_n, uu_n) = a(uu_n, uv_n) \le C \uu_n\ \, \uv_n\.
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:
Proof
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.

\uu_n\ \le \frac{C}{c} \inf_{v_n\in V_n} \uv_n\.
The error uu_n between the original and the Galerkin solution admits the estimate
Quasibest approximation (Céa's lemma)
Since V_n \subset V, boundedness and ellipticity of the bilinear form apply to V_n. Therefore, the wellposedness of the Galerkin problem is actually inherited from the wellposedness of the original problem.
Wellposedness of the Galerkin equation
By the LaxMilgram theorem (see weak formulation), these two conditions imply wellposedness 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).

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.
The analysis will mostly rest on two properties of the bilinear form, namely
The analysis of these methods proceeds in two steps. First, we will show that the Galerkin equation is a wellposed 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.
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.

a(u,v) = a(v,u).
Here, we will restrict ourselves to symmetric bilinear forms, that is
Analysis of Galerkin methods
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.
Symmetry of the matrix

A_{ij} = a(e_j, e_i), \quad f_i = f(e_i).
This previous equation is actually a linear system of equations Au=f, where

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.
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(u_n, e_i) = f(e_i) \quad i=1,\ldots,n.
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
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.
Matrix form

a(\epsilon_n, v_n) = a(u,v_n)  a(u_n, v_n) = f(v_n)  f(v_n) = 0.
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 = uu_n which is the error between the solution of the original problem, u, and the solution of the Galerkin equation, u_n
Galerkin orthogonality
We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finitedimensional vector subspace allows us to numerically compute u_n as a finite linear combination of the basis vectors in V_n .

Find u_n\in V_n such that for all v_n\in V_n, a(u_n,v_n) = f(v_n).
Choose a subspace V_n \subset V of dimension n and solve the projected problem:
Galerkin Dimension Reduction
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.

find u\in V such that for all v\in V, a(u,v) = f(v).
Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space V, namely,
A problem in weak formulation
Introduction with an abstract problem
Contents

Introduction with an abstract problem 1

A problem in weak formulation 1.1

Galerkin Dimension Reduction 1.2

Galerkin orthogonality 1.3

Matrix form 1.4

Symmetry of the matrix 1.4.1

Analysis of Galerkin methods 2

Wellposedness of the Galerkin equation 2.1

Quasibest approximation (Céa's lemma) 2.2

References 3

External links 4
Examples of Galerkin methods are:
).
Walther Ritz (after ^{[4]}Ritz–Galerkin method) or ^{[3]}[2]
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