Abstract: We present an original preconditioner to solve the discretized Poisson equation with the conjugate gradient method. The preconditioner is based on the approximation of the inverse Poisson operator. In particular, we construct the inverse operator corresponding to a layered model which is close to the original one. The action of the preconditioner at each iteration requires the system of equations to be solved corresponding to the layered media. It is done by applying the spectral decomposition of the matrix corresponding to the 1D problem (spectral decomposition along one spatial direction) and further direct solution of a series of 1D problems along the other direction. We provide the numerical and theoretical study of the convergence rate depending on the way the layered model is constructed to approximate the original. We consider four cases: maximal value over the layer, minimal value, arithmetic averaging, and harmonic averaging. For the first two cases, we prove analytically that the convergence rate of the preconditioned conjugate gradient method is independent of the size of the problem but depends on the coefficient contrast. For the other two cases, the weak dependence on the size of the problem is illustrated numerically.

Cite: Manaev A. , Lisitsa V.
Eigenvalue decomposition-based preconditioner to solve Poisson equation in application to digital rock physics
Applied Numerical Mathematics. 2026. V.230. P.227-248. DOI: 10.1016/j.apnum.2026.07.015 Scopus OpenAlex

Dates:

Submitted:	Jun 20, 2025
Accepted:	Jul 31, 2026
Published online:	Aug 5, 2026

Identifiers:

≡ Scopus:	2-s2.0-105046568740
≡ OpenAlex:	W7196936660

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