Petrov–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 ${\displaystyle V}$ and ${\displaystyle W}$, namely,

find ${\displaystyle u\in V}$ such that ${\displaystyle a(u,w)=f(w)}$ for all ${\displaystyle w\in W}$.

Here, ${\displaystyle a(\cdot ,\cdot )}$ is a bilinear form and ${\displaystyle f}$ is a bounded linear functional on ${\displaystyle W}$.

Petrov-Galerkin dimension reduction

Choose subspaces ${\displaystyle V_n\subset V}$ of dimension n and ${\displaystyle W_m\subset W}$ of dimension m and solve the projected problem:

Find ${\displaystyle v_n\in V_n}$ such that ${\displaystyle a(v_n,w_m)=f(w_m)}$ for all ${\displaystyle w_m\in W_m}$.

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 ${\displaystyle v_n}$ as a finite linear combination of the basis vectors in ${\displaystyle V_n}$.

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 ${\displaystyle W_m\subset W}$, we can use ${\displaystyle w_m}$ as a test vector in the original equation. Subtracting the two, we get the relation for the error, ${\displaystyle {ϵ}_n=v-v_n}$ which is the error between the solution of the original problem, ${\displaystyle v}$, and the solution of the Galerkin equation, ${\displaystyle v_n}$, as follows

${\displaystyle a({ϵ}_n,w_m)=a(v,w_m)-a(v_n,w_m)=f(w_m)-f(w_m)=0}$ for all ${\displaystyle w_m\in W_m}$.

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 ${\displaystyle v^1,v^2,\dots ,v^n}$ be a basis for ${\displaystyle V_n}$ and ${\displaystyle w^1,w^2,\dots ,w^m}$ be a basis for ${\displaystyle W_m}$. Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find ${\displaystyle v_n\in V_n}$ such that

${\displaystyle a(v_n,w^j)=f(w^j)j=1,\dots ,m.}$

We expand ${\displaystyle v_n}$ with respect to the solution basis, ${\displaystyle v_n=\sum _{i=1}^n{x}^i{v}^i}$ and insert it into the equation above, to obtain

${\displaystyle a\left(\sum _{i=1}^n{x}^i{v}^i,w^j\right)=\sum _{i=1}^n{x}^{i}a(v^i,w^j)=f(w^j)j=1,\dots ,m.}$

This previous equation is actually a linear system of equations ${\displaystyle A^{T}x=f}$, where

${\displaystyle A_{ij}=a(v^i,w^j),f_j=f(w^j).}$

Symmetry of the matrix

Due to the definition of the matrix entries, the matrix ${\displaystyle A}$ is symmetric if ${\displaystyle V=W}$, the bilinear form ${\displaystyle a(\cdot ,\cdot )}$ is symmetric, ${\displaystyle n=m}$, ${\displaystyle V_n=W_m}$, and ${\displaystyle v^i=w^j}$ for all ${\displaystyle i=j=1,\dots ,n=m.}$ In contrast to the case of Bubnov-Galerkin method, the system matrix ${\displaystyle A}$ is not even square, if ${\displaystyle n\ne m.}$