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.
[2] Petrov-Galerkin's method is a natural extension of Galerkin method and can be similarly introduced as follows.
Let us consider an abstract problem posed as a weak formulation on a pair of Hilbert spaces
is a bilinear form and
is a bounded linear functional on
Choose subspaces
of dimension m and solve the projected problem: 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
The key property of the Petrov-Galerkin approach is that the error is in some sense "orthogonal" to the chosen subspaces.
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 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.
be a basis for
be a basis for
Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find
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 Due to the definition of the matrix entries, the matrix
is symmetric if
, the bilinear form
is symmetric,
In contrast to the case of Bubnov-Galerkin method, the system matrix