Convergence analysis of two-grid methods for symmetric positive semidefinite systems

Two-grid theory plays a fundamental role in the design and analysis of multigrid methods. This paper is devoted to a new convergence analysis of two-grid methods for singular and symmetric positive semidefinite systems. Specifically, we derive a concise identity for characterizing the convergence factor of two-grid methods, with the Moore--Penrose inverse of coarse-grid matrix being used as a coarse solver. Furthermore, we present a convergence estimate for two-grid methods with approximate coarse solvers. Our new theory does not require any additional assumptions on the coefficient matrix, especially on its null space.

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