An efficient Galerkin method for problems with physically realistic boundary conditions

The Galerkin method is often employed for numerical integration of evolutionary equations, such as the Navier–Stokes equation or the magnetic induction equation. Application of the method requires solving at each time step a linear equation of the form $P(Av-f)=0$, where v is an element of a finite-dimensional space $V$ with a basis satisfying the boundary conditions. We propose an algorithm giving an opportunity to reduce the computational cost for such a problem. Suppose there exists a space $W$ that contains $V$, the difference between the dimensions of $W$ and $V$ is small compared to the dimension of $V$, and solving the problem $P(Aw-f)=0$, where w is an element of $W$, requires less operations than solving the original problem. The solution to $P(Av-f)=0$ is found in two steps: we solve the problem $P(Aw-f)=0$ in $W$ and compute a correction $q=v-w$ that belongs to the kernel of PA, which is a complement to $V$ in $W$; q is computed using a basis in the orthogonal complement to $V$ in $W$. We discuss the algorithm both in the general form and its instance when $W$ is spanned by Chebyshev polynomials.