A new convergence analysis of two-level hierarchical basis methods

This paper is concerned with the convergence analysis of two-level hierarchical basis (TLHB) methods in a general setting, where the decomposition associated with two hierarchical component spaces is not required to be a direct sum. The TLHB scheme can be regarded as a combination of compatible relaxation and coarse-grid correction. Most of the previous works focus on the case of exact coarse solver, and the existing identity for the convergence factor of exact TLHB methods involves a tricky max-min problem. In this work, we present a new and purely algebraic analysis of TLHB methods, which gives a succinct identity for the convergence factor of exact TLHB methods. The new identity can be conveniently utilized to derive an optimal interpolation and analyze the influence of coarse space on the convergence factor. Moreover, we establish two-sided bounds for the convergence factor of TLHB methods with inexact coarse solver, which extend the existing TLHB theory.