Fully discrete Heterogeneous Multiscale Method for parabolic problems with multiple spatial and temporal scales

TL;DR

This paper develops a fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scales, providing a new approach and rigorous error analysis validated by numerical experiments.

Contribution

It introduces an alternative cell problem formulation and offers a detailed a priori error analysis for the fully discretized multiscale method.

Findings

The method achieves theoretical convergence rates.

Numerical experiments confirm the error estimates.

The approach simplifies the cell problem setup.

Abstract

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.