A space-time multiscale method for parabolic problems

TL;DR

This paper introduces a space-time multiscale method for parabolic problems with highly oscillatory coefficients, enabling efficient and accurate solutions through localized correctors and a variational multiscale framework.

Contribution

It develops a novel space-time multiscale approach based on the Variational Multiscale Method, with localized correctors for efficient computation independent of oscillation scales.

Findings

First-order convergence proven regardless of coefficient oscillations.

Space-time correctors decay exponentially, enabling localization.

Numerical examples demonstrate method efficiency and accuracy.

Abstract

We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the Variational Multiscale Method in the context of a space-time formulation and computes a coarse-scale representation of the differential operator that is enriched by auxiliary space-time corrector functions. Once computed, the coarse-scale representation allows us to efficiently obtain well-approximating discrete solutions for multiple right-hand sides. We prove first-order convergence independently of the oscillation scales in the coefficient and illustrate how the space-time correctors decay exponentially in both space and time, making it possible to localize the corresponding computations. This localization allows us to define a practical and computationally efficient method in terms of complexity and memory, for which we provide a posteriori error estimates and present numerical examples.