Petrov–Galerkin method
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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
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
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
Notes
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