World Library  
Flag as Inappropriate
Email this Article

Galerkin method

Article Id: WHEBN0002664839
Reproduction Date:

Title: Galerkin method  
Author: World Heritage Encyclopedia
Language: English
Subject: Finite element method, Missing science topics/ExistingMathG, Galerkin, Ted Belytschko, EFG
Collection: Articles Containing Proofs, Numerical Analysis, Numerical Differential Equations
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Galerkin method

In


  • Hazewinkel, Michiel, ed. (2001), "Galerkin method",  
  • Galerkin's Method
  • Galerkin Method from MathWorld

External links

  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

References

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

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\|.

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.

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

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

Quasi-best approximation (Céa's lemma)

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.

Well-posedness of the Galerkin equation

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).

  • 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 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.

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 = u-u_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 finite-dimensional 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
    • Well-posedness of the Galerkin equation 2.1
    • Quasi-best approximation (Céa's lemma) 2.2
      • Proof 2.2.1
  • References 3
  • External links 4

Examples of Galerkin methods are:

). Walther Ritz (after [4]Ritz–Galerkin method) or [3][2]

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.