We present numerical algorithm to estimate the formation factor of porous materials using the micro-tomographic images. The key part of the algorithm is the numerical solution of the 3D Poisson equation with rapidly varying high-contrast coefficients. The suggested algorithm is based on the preconditioned Conjugate Gradient method. The preconditioner is constructed as the inverse Laplace operator corresponding to a homogeneous model. It can be inverted using the spectral decomposition of two tridiagonal matrices corresponding to the approximation of 1D derivatives. The resulting series of 1D problems is solved by Thomas algorithm. We prove analytically and illustrate numerically that the condition number, and thus the convergence rate of the preconditioned problem depends on the contrast of the equation coefficients, but it is independent on the problems size. We illustrate that the preconditioner can be efficiently applied to the original problem with rapidly varying high-contrast coefficients and to the statement where the solution is computed only in the pore space. The algorithm is implemented using Graphic Processor Units. The use of modern GPUs allows us to solve problems of up to size 1 0 9 with a single unit. • Solving Poisson equation in complex models of porous materials. • Use of CG with an inverse homogeneous Laplace operator as a preconditioner. • Solving problems of the size of 1 0 9 in several minutes with a single GPU.
Lisitsa et al. (2026) studied this question.