A Priori Error Bounds for the Approximate Deconvolution Leray Reduced Order Model

Abstract

The approximate deconvolution Leray reduced order model (ADL-ROM) uses spatial filtering to increase the ROM stability, and approximate deconvolution to increase the ROM accuracy. In the under-resolved numerical simulation of convection-dominated flows, ADL-ROM was shown to be significantly more stable than the standard ROM, and more accurate than the Leray ROM. In this paper, we prove a priori error bounds for the approximate deconvolution operator and ADL-ROM. To our knowledge, these are the first numerical analysis results for approximate deconvolution in a ROM context. We illustrate these numerical analysis results in the numerical simulation of convection-dominated flows.

Figures (4)

• Figure 1: Approximate deconvolution rates of convergence for (a) $r=99$ and (b) $r=100$.
• Figure 1: ADL-ROM approximation error $E_{L^{2}}^{ADL-ROM}$ for decreasing $\delta$ values.
• Figure 2: ADL-ROM approximation error $E_{L^{2}}^{ADL-ROM}$ for increasing $r$ values.
• Figure 3: Linear regression of $E_{L^{2}}^{ADL-ROM}$ with respect to $\Lambda^{r}_{H^1}$.

Theorems & Definitions (23)

• Lemma 2.1
• Lemma 2.2: Discrete Gronwall Lemma
• Definition 3.1: ROM Projection
• Lemma 3.1
• Remark 3.1
• Definition 3.2: Continuous Differential Filter
• Definition 3.3: ROM Differential Filter
• Lemma 3.2
• Lemma 3.3
• Remark 3.2