Petrov–Galerkin method explained

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

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

A problem in weak formulation

V and W , namely,

find

u\in V such that a(u,w) = f(w) for all w\in W .

Here,

a( ⋅ , ⋅ ) is a bilinear form and f is a bounded linear functional on W .

Petrov-Galerkin dimension reduction

Choose subspaces

V _{n} \subset V of dimension n and W _{m} \subset W of dimension m and solve the projected problem:

Find

v _{n\in} V _{n} such that a(v _{n,w} _{m)} = f(w _{m)} for all w _{m\in} W _{m} .

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

v _{n} as a finite linear combination of the basis vectors in V _{n} .

Petrov-Galerkin generalized orthogonality

The key property of the Petrov-Galerkin approach is that the error is in some sense "orthogonal" to the chosen subspaces. Since

W _{m} \subset W , we can use w _{m} as a test vector in the original equation. Subtracting the two, we get the relation for the error, \epsilon _{n} = v - v _{n} which is the error between the solution of the original problem, v , and the solution of the Galerkin equation, v _{n} , as followsa(\epsilon _{n,} w _{m)} = a(v,w _{m)} - a(v _{n,} w _{m)} = f(w _{m)} - f(w _{m)} = 0 for all w _{m\in} W _{m} .

Matrix form

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

v ^{1,} v ^{2,\ldots,} v ^{n} be a basis for V _{n} and w ^{1,} w ^{2,\ldots,} w ^{m} be a basis for W _{m} . Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find v _{n} \in V _{n} such thata(v _{n,} w ^{j)} = f(w ^{j)} j =1 ,\ldots, m.

We expand

v _{n} with respect to the solution basis, v _{n} = \sum ^{n} _{i =1} x ^{i} v ^{i} and insert it into the equation above, to obtaina\left(\sum ^{n} _{i =1} x ^{i} v ^{i,} w ^{j\right)} = \sum ^{n} _{i =1} x ^{i} a(v ^{i,} w ^{j)} = f(w ^{j)} j =1 ,\ldots,m.

This previous equation is actually a linear system of equations

A ^{Tx=f} , whereA _{ij} = a(v ^{i,} w ^{j),} f _{j} = f(w ^{j).}

Symmetry of the matrix

Due to the definition of the matrix entries, the matrix

A is symmetric if V = W , the bilinear form a( ⋅ , ⋅ ) is symmetric, n = m , V _{n=W} _{m} , and v ^{i=w} ^{j} for all i = j =1 ,\ldots, n = m. In contrast to the case of Bubnov-Galerkin method, the system matrix A is not even square, if n ≠ m.

See also

• Bubnov-Galerkin method

Notes and References

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