Optimal higher-order convergence rates for parabolic multiscale problems
For researchers in numerical analysis and multiscale simulation, this work solves the convergence rate degradation issue in higher-order LOD methods for time-dependent problems.
This paper introduces a higher-order multiscale method for parabolic problems with oscillatory coefficients, achieving optimal convergence rates without additional assumptions on the coefficients. Numerical experiments confirm the theoretical error estimates.
In this paper, we introduce a higher-order multiscale method for time-dependent problems with highly oscillatory coefficients. Building on the localized orthogonal decomposition (LOD) framework, we construct enriched correction operators to enrich the multiscale spaces, ensuring higher-order convergence without requiring assumptions on the coefficient beyond boundedness. This approach addresses the challenge of a reduction of convergence rates when applying higher-order LOD methods to time-dependent problems. Addressing a parabolic equation as a model problem, we prove the exponential decay of these enriched corrections and establish rigorous a priori error estimates. Numerical experiments confirm our theoretical results.