Nested BDDC for a saddle-point problem

TL;DR

This paper introduces a recursive Nested BDDC method for saddle-point problems that efficiently solves for flux and pressure variables by reusing components across multiple hierarchical levels, reducing computational costs.

Contribution

The paper presents a novel recursive Nested BDDC algorithm that solves saddle-point problems efficiently by reusing components and approximating coarse problems across multiple levels.

Findings

Condition number bound established

Numerical experiments confirm theoretical efficiency

Hierarchical approach reduces computational cost

Abstract

We propose a Nested BDDC for a class of saddle-point problems. The method solves for both flux and pressure variables. The fluxes are resolved in three-steps: the coarse solve is followed by subdomain solves, and last we look for a divergence-free flux correction and pressure variables using conjugate gradients with a Multilevel BDDC preconditioner. Because the coarse solve in the first step has the same structure as the original problem, we can use this procedure recursively and solve (a hierarchy of) coarse problems only approximately, utilizing the coarse problems known from the BDDC. The resulting algorithm thus first performs several upscaling steps, and then solves a hierarchy of problems that have the same structure but increase in size while sweeping down the levels, using the same components in the first and in the third step on each level, and also reusing the components from the higher levels. Because the coarsening can be quite aggressive, the number of levels can be kept small and the additional computational cost is significantly reduced due to the reuse of the components. We also provide the condition number bound and numerical experiments confirming the theory.