BDDC and FETI-DP under Minimalist Assumptions

TL;DR

This paper presents a simple, algebraic framework for FETI-DP, BDDC, and P-FETI-DP preconditioners, showing their equivalence and properties under minimal assumptions without relying on specific substructure definitions.

Contribution

It introduces a minimalist algebraic framework that unifies FETI-DP and BDDC, demonstrating their equivalence and extending condition number bounds.

Findings

FETI-DP and BDDC are algebraically equivalent.

Standard condition number bounds apply to the abstract operators.

The framework does not depend on specific substructure or coarse degrees of freedom.

Abstract

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.