Multispace and Multilevel BDDC

TL;DR

This paper introduces a new family of Multispace BDDC methods for large linear systems from elliptic PDEs, providing theoretical bounds and numerical validation for multilevel recursive approaches.

Contribution

It formulates a novel abstract Multispace BDDC framework and derives condition number bounds, extending BDDC to multilevel recursive schemes with theoretical guarantees.

Findings

Polylogarithmic condition number bounds for fixed multilevel levels

Recursive BDDC can be applied approximately to coarse problems

Numerical experiments confirm theoretical bounds

Abstract

BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.