BDDC and FETI-DP algorithms with adaptive coarse spaces for three-dimensional elliptic problems with oscillatory and high contrast coefficients

TL;DR

This paper introduces adaptive BDDC and FETI-DP algorithms with enriched coarse spaces for 3D elliptic problems with oscillatory and high contrast coefficients, improving robustness and condition number bounds.

Contribution

It develops a novel adaptive coarse space construction using generalized eigenvalue problems, enhancing preconditioner robustness for complex 3D elliptic problems.

Findings

Condition number bound proportional to $\lambda_{TOL}$

Numerical tests confirm robustness of the method

Coarse space enrichment improves solver efficiency

Abstract

BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance $\lambda_{TOL}$. Upper bounds for condition numbers of the preconditioned systems are shown to be $C \lambda_{TOL}$, with the constant $C$ depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.