On the convergence of an extrapolation cascadic multigrid method for elliptic problems

Classical cascadic multigrid method is optimal under the energy norm for H2-regular elliptic problems. This paper analyzes an extrapolation cascadic multigrid (EXCMG) method, originally proposed by Chen et al. (2008) for solving second-order elliptic equations. Following the idea of Bornemann and Deuflhard (1996), we present a superconvergence result for the EXCMG method, which enables us to show that the EXCMG method with the conjugate gradient method as a basic iterative scheme is optimal for H3-regular elliptic problems in three dimension with respect to the L2-norm. Moreover, we also prove the super-optimality of the EXCMG method under the energy norm for H2+α-regular (0<α≤1) problems in both two and three dimensions with a reasonable assumption on asymptotic error expansions. Finally, numerical results are presented to verify our theoretical analysis.