Multilevel Additive Schwarz Preconditioners for High‐Order Block Implicit Methods for Parabolic Equations

ABSTRACT Block implicit method (BIM) is a family of time discretization schemes with desirable stability properties, high order of accuracy, and requires a single initial value. Moreover, BIM is also time‐parallel since the solutions at several time steps can be computed simultaneously at a time. However, its resulting linear system is often large and highly ill‐conditioned. In this paper, we present a multilevel additive Schwarz preconditioner for solving the large sparse system of algebraic equations arising from the discretization by BIM in time and finite element in space. Under some mild assumption, we prove an optimal convergence theory showing that the convergence rate is bounded independently of the spatial mesh sizes, the time step size, the number of subdomains, and the number of levels. Numerical experiments carried out on a parallel computer with thousands of processors confirm the optimality and scalability of the method, which performs better than the low‐order schemes in terms of the compute time.