Strang-Type Preconditioners for Differential-Algebraic Equations

TLDR: It was shown that the preconditioners are nonsingular when the BVM is A?, µ-?-stable, and the eigenvalues of preconditionsed matrices are clustered, so the number of iterations for solving the precONDitioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh.

Abstract: We consider linear constant coefficient differential-algebraic equations (DAEs) Ax?(t) + Bx(t) = f(t) where A, B are square matrices and A is singular. If det(?A + B) with ? ? C is not identically zero, the system of DAEs is solvable and can be separated into two uncoupled subsystems. One of them can be solved analytically and the other one is a system of ordinary differential equations (ODEs). We discretize the ODEs by boundary value methods (BVMs) and solve the linear system by using the generalized minimal residual (GMRES) method with Strang-type block-circulant preconditioners. It was shown that the preconditioners are nonsingular when the BVM is A?, µ-?-stable, and the eigenvalues of preconditioned matrices are clustered. Therefore, the number of iterations for solving the preconditioned systems by the GMRES method is bounded by a constant that is independent of the discretization mesh. Numerical results are also given.