Optimal higher-order convergence rates for parabolic multiscale problems

TL;DR

This work tackles the challenge of obtaining higher-order spatial convergence for parabolic problems with highly oscillatory diffusion by introducing enriched corrections to the p-LOD framework. The authors prove exponential decay of the enriched corrections, derive a rigorous semi-discrete error bound independent of the oscillation scale $\varepsilon$, and show that optimal convergence $\|e\|_{H^1} \sim H^{p+2}$ is achievable without additional spatial regularity assumptions, by selecting $j=\lceil p/2\rceil$ and $\ell\sim C_p|\log H|$. A practical localization strategy enables computing all enriched corrections on a common patch, maintaining sparsity and efficiency. Numerical experiments in 1D and 2D with oscillatory and random coefficients confirm the theory and demonstrate the method’s potential for efficient multiscale simulations of time-dependent diffusion problems.

Abstract

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.

Figures (4)

• Figure 2.1: Illustration of the element $K$ with patch $\Nb[K][3]$ and element $G_1$ with patch $\Nb[G_1][1]$. Here we have $\mathop{\mathrm{dist}}\limits(G_1,K)=\mu = 2$.
• Figure 4.1: Coefficient with some coarse structure and fine oscillations used for \ref{['ex:example1']} (left) and random diffusion coefficient for \ref{['ex:example2']} (right).
• Figure 4.2: Relative energy errors for \ref{['ex:example1']} with respect to the mesh size (left) and degrees of freedom (right). The legend applies to both plots.
• Figure 4.3: Relative energy errors for \ref{['ex:example2']} with random diffusion coefficient and $p=3$ with different localization parameters (left). Localization errors for \ref{['ex:example3']} of a one-dimensional random diffusion coefficient (right). On the right, the first three lines at the top are the classical higher-order LOD with $j=0$, and the lines below that are $j=1$ for $p=2$ and then $j=2$ for $p=3,4$

Theorems & Definitions (26)

• Remark 2.2: Regularity
• Lemma 2.3: DonHM23
• Remark 2.4: global corrections
• Lemma 2.5: DonHM23
• Lemma 2.6: DonHM23
• Lemma 2.7: DonHM23
• Lemma 2.8: Error of the ideal enriched correction
• proof
• Remark 2.9
• Lemma 2.10