Computational homogenization of parabolic equations with memory effects for a periodic heterogeneous medium

TL;DR

This paper develops a computational homogenization framework for parabolic equations with memory effects in periodic heterogeneous media, enabling efficient numerical solutions of nonlocal diffusion models with stability guarantees.

Contribution

It introduces a novel computational approach for homogenizing nonstationary processes with memory effects, transforming nonlocal problems into local ones for easier numerical treatment.

Findings

Effective diffusion tensor computed via finite element methods.

Memory kernel approximated by exponential sums from spectral problems.

Unconditional stability proved for the discretized schemes.

Abstract

In homogenization theory, mathematical models at the macro level are constructed based on the solution of auxiliary cell problems at the micro level within a single periodicity cell. These problems are formulated using asymptotic expansions of the solution with respect to a small parameter, which represents the characteristic size of spatial heterogeneity. When studying diffusion equations with contrasting coefficients, special attention is given to nonlocal models with weakly conducting inclusions. In this case, macro-level processes are described by integro-differential equations, where the difference kernel is determined by the solution of a nonstationary cell problem. The main contribution of this work is the development of a computational framework for the homogenization of nonstationary processes, accounting for memory effects. The effective diffusion tensor is computed using a standard numerical procedure based on finite element discretization in space. The memory kernel is approximated by a sum of exponentials obtained from solving a partial spectral problem on the periodicity cell. The nonlocal macro-level problem is transformed into a local one, where memory effects are incorporated through the solution of auxiliary nonstationary problems. Standard two-level time discretization schemes are employed, and unconditional stability of the discrete solutions is proved in appropriate norms. Key aspects of the proposed computational homogenization technique are illustrated by solving a two-dimensional model problem.