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.J. N. Reddy: An introduction to the finite element method, 2006, Mcgraw–Hill 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."Georgii Ivanovich Petrov (on his 100th birthday)", Fluid Dynamics, May 2012, Volume 47, Issue 3, pp 289-291, DOI 10.1134/S0015462812030015
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 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 map and is a bounded linear functional on .
Petrov-Galerkin dimension reduction 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 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 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 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 Bubnov-Galerkin method
Notes