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. 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.
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 V {\displaystyle V} and W {\displaystyle W} , namely,
find u ∈ V {\displaystyle u\in V} such that a ( u , w ) = f ( w ) {\displaystyle a(u,w)=f(w)} for all w ∈ W {\displaystyle w\in W} . Here, a ( ⋅ , ⋅ ) {\displaystyle a(\cdot ,\cdot )} is a bilinear map and f {\displaystyle f} is a bounded linear functional on W {\displaystyle W} .
Petrov-Galerkin dimension reduction Choose subspaces V n ⊂ V {\displaystyle V_{n}\subset V} of dimension n and W m ⊂ W {\displaystyle W_{m}\subset W} of dimension m and solve the projected problem:
Find v n ∈ V n {\displaystyle v_{n}\in V_{n}} such that a ( v n , w m ) = f ( w m ) {\displaystyle a(v_{n},w_{m})=f(w_{m})} for all w m ∈ W m {\displaystyle 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 {\displaystyle v_{n}} as a finite linear combination of the basis vectors in V n {\displaystyle 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 ⊂ W {\displaystyle W_{m}\subset W} , we can use w m {\displaystyle w_{m}} as a test vector in the original equation. Subtracting the two, we get the relation for the error, ϵ n = v − v n {\displaystyle \epsilon _{n}=v-v_{n}} which is the error between the solution of the original problem, v {\displaystyle v} , and the solution of the Galerkin equation, v n {\displaystyle v_{n}} , as follows
a ( ϵ n , w m ) = a ( v , w m ) − a ( v n , w m ) = f ( w m ) − f ( w m ) = 0 {\displaystyle a(\epsilon _{n},w_{m})=a(v,w_{m})-a(v_{n},w_{m})=f(w_{m})-f(w_{m})=0} for all w m ∈ W m {\displaystyle w_{m}\in W_{m}} .
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