Petrov–Galerkin method

(Redirected from Petrov-Galerkin methods)

The Petrov–Galerkin method is a mathematical method used to approximate solutions of partial differential equations which contain terms with odd order and where the test function and solution function belong to different function spaces.[1] It can be viewed as an extension of Bubnov-Galerkin method where the bases of test functions and solution functions are the same. In an operator formulation of the differential equation, Petrov–Galerkin method can be viewed as applying a projection that is not necessarily orthogonal, in contrast to Bubnov-Galerkin method.

It is named after the Soviet scientists Georgy I. Petrov and Boris G. Galerkin.[2]

Introduction with an abstract problem

edit

Petrov-Galerkin's method is a natural extension of Galerkin method and can be similarly introduced as follows.

A problem in weak formulation

edit

Let us consider an abstract problem posed as a weak formulation on a pair of Hilbert spaces   and  , namely,

find   such that   for all  .

Here,   is a bilinear form and   is a bounded linear functional on  .

Petrov-Galerkin dimension reduction

edit

Choose subspaces   of dimension n and   of dimension m and solve the projected problem:

Find   such that   for all  .

We 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   as a finite linear combination of the basis vectors in  .

Petrov-Galerkin generalized orthogonality

edit

The key property of the Petrov-Galerkin approach is that the error is in some sense "orthogonal" to the chosen subspaces. Since  , we can use   as a test vector in the original equation. Subtracting the two, we get the relation for the error,   which is the error between the solution of the original problem,  , and the solution of the Galerkin equation,  , as follows

  for all  .

Matrix form

edit

Since the aim of the approximation is producing a linear system of equations, we build its matrix form, which can be used to compute the solution algorithmically.

Let   be a basis for   and   be a basis for  . Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find   such that

 

We expand   with respect to the solution basis,   and insert it into the equation above, to obtain

 

This previous equation is actually a linear system of equations  , where

 

Symmetry of the matrix

edit

Due to the definition of the matrix entries, the matrix   is symmetric if  , the bilinear form   is symmetric,  ,  , and   for all   In contrast to the case of Bubnov-Galerkin method, the system matrix   is not even square, if  

See also

edit

Notes

edit
  1. ^ J. N. Reddy: An introduction to the finite element method, 2006, Mcgraw–Hill
  2. ^ "Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015