Preconditioners for Saddle Point Problems on Truncated Domains in Phase Separation Modelling

TL;DR

This paper develops and analyzes preconditioners for solving linear saddle point problems arising from phase separation models, specifically addressing truncated domains in the discretized Cahn-Hilliard equation with obstacle potential.

Contribution

The paper introduces and evaluates three preconditioners for truncated saddle point problems, providing eigenvalue bounds and demonstrating their optimality through numerical experiments.

Findings

Eigenvalue bounds for preconditioned truncated systems

Preconditioners maintain bounded eigenvalues similar to untruncated systems

Numerical results confirm the effectiveness and optimality of the proposed solvers

Abstract

The discretization of Cahn-Hilliard equation with obstacle potential leads to a block 2 by 2 non-linear system, where the p1, 1q block has a non-linear and non-smooth term. Recently a globally convergent Newton Schur method was proposed for the non-linear Schur complement corresponding to this non-linear system. The solver may be seen as an inexact Uzawa method which has the falvour of an active set method in the sense that the active sets are first identified by solving a quadratic obstacle problem corresponding to the p1, 1q block of the block 2 by 2 nonlinear system, and a new decent direction is obtained after discarding the active set region. The problem becomes linear on nonactive set, and corresponds to solving a linear saddle point problem on truncated domains. For solving the quadratic obstacle problem, various optimal multigrid like methods have been proposed. In this paper solvers for the truncated saddle point problem is considered. Three preconditioners are considered, two of them have block diagonal structure, and the third one has block tridiagonal structure. One of the block diagonal preconditioners is obtained by adding certain scaling of stiffness and mass matrices, whereas, the remaining two involves Schur complement. Eigenvalue bound and condition number estimates are derived for the preconditioned untruncated problem. It is shown that the extreme eigenvalues of the preconditioned truncated system remain bounded by the extreme eigenvalues of the preconditioned untruncated system. Numerical experiments confirm the optimality of the solvers.