A family of physics-based preconditioners for solving elliptic equations on highly heterogeneous media

Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials, geological rock properties and thermal and electrical conductivity. The main objective of this work is to construct a method as algebraic as possible. However, the underlying physics is utilized to distinguish between high and low degrees of freedom which is central to the construction of the proposed preconditioner. Namely, we propose an algebraic way of separating binary-like systems according to a given threshold into high- and low-conductivity regimes of coefficient size O(m) and O(1), respectively where m≫1. So, the proposed preconditioner is essentially physics-based because without the utilization of underlying physics such an algebraic distinction, hence, the construction of the preconditioner would not be possible. The condition number of the linear system depends both on the mesh size Δx and the coefficient size m. For our purposes, we address only the m dependence since the condition number of the linear system is mainly governed by the high-conductivity sub-block. Thus, the proposed strategy is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a two-stage preconditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation strategy (if ever needed). Due to its algebraic nature, the proposed approach can support a wide range of realistic geometries (e.g., layered and channelized media). Numerical examples show that the proposed class of physics-based preconditioners are more effective and robust compared to a class of Krylov-based deflation methods on highly heterogeneous media.