The paper «An efficient Galerkin method for problems with physically realistic boundary» conditions was published in the journal «Computer Physics Communications». The author of the article is doctor of physical and mathematical sciences O.M.Podvigina.
We propose a new algorithm to solve an equation of the form P(Ax-b)=0 emerging in the course of numerical integration of the equations of hydro- and magnetodynamics, such as the Navier-Stokes equation or the magnetic induction equation, by the Galerkin method. Here x is an element of a finite-dimensional space V with a basis, that satisfies boundary conditions, P – is the projection on this space and A is a linear operator. Usually the coefficients of x decomposed in the basis are found by calculating the matrix of PA acting on V and solving the respective system of linear equations. In case of physically realistic boundary conditions (such as no-slip for the velocity or insulating boundary conditions for the magnetic field) the basis is not orthogonal and the solution of the problem might be computationally demanding. We propose a method that allows to reduce the computational cost for such a problem.
Suppose, that there exists a space W that contains V and the difference between the dimensions of W and V is small, compare to the dimension of V. Also, the solution of the problem P(Ay-b)=0, where y is an element of W, requires less operations than the solution of the original problem. The equation P(Ax-b)=0 is solved in two steps: first we solve the problem P(Ay-b)=0, where y is an element of W, and second we evaluate the correction x-y, that belongs to a complement to V in W. Since the dimension of the complement is small the proposed algorithm is more efficient than the traditional one. The algorithm is applied to study the generation of magnetic field by convective flows.
