Solution of parameter-dependent diffusion equation in layered media

TL;DR

The paper tackles parameter-dependent diffusion in a 2D layered domain with layer-wise constants $y_i$, aiming to construct compact parameter-to-solution maps. It derives a closed-form three-term representation for the two-layer case and extends the idea to $N$ layers via a slow-fast decomposition of the interface space, yielding $u(y)\approx \sum_i \frac{1}{y_i} w_{\Omega_i}+\sum_i \frac{2}{y_i+y_{i+1}} w_{fi}+u_s(y)$ with $u_s(y)$ in a low-dimensional subspace; it provides rigorous error estimates, harmonic-extension tools, and finite element realizations, plus exponential decay bounds for the approximation error and Kolmogorov $n$-width results. The framework clarifies why reduced-basis and sampling-based methods perform well for such parametric diffusion problems and offers practical constructions (continuous and FE) to build efficient solvers for multi-layer geometries. Numerical experiments corroborate exponential convergence and demonstrate that a small slow-space is sufficient to achieve high accuracy. Overall, the work advances theoretically grounded, low-rank representations for parameter-dependent diffusion in layered media with potential impact on efficient solvers and uncertainty quantification.

Abstract

This work studies the parameter-dependent diffusion equation in a two-dimensional domain consisting of locally mirror symmetric layers. It is assumed that the diffusion coefficient is a constant in each layer. The goal is to find approximate parameter-to-solution maps that have a small number of terms. It is shown that in the case of two layers one can find a solution formula consisting of three terms with explicit dependencies on the diffusion coefficient. The formula is based on decomposing the solution into orthogonal parts related to both of the layers and the interface between them. This formula is then expanded to an approximate one for the multi-layer case. We give an analytical formula for square layers and use the finite element formulation for more general layers. The results are illustrated with numerical examples and have applications for reduced basis methods by analyzing the Kolmogorov n-width.

Figures (4)

• Figure 1: An example of a layered geometry with $N=3$. The diffusion coefficient is a constant $y_i$ in each subdomain $\Omega_i$.
• Figure 2: A locally mirror symmetric finite element mesh for the $3\times 1$ square grid. Standard uniform order finite element spaces associated to this mesh satisfy Assumption \ref{['ass:symm']}.
• Figure 3: The different components of the solution and the error for the geometry of Fig. \ref{['fig:geom_crown']} with $f=x_2$ and $y=[2,4,3]$. The sum of the intermediate solutions \ref{['fig:domain_sols']}, \ref{['fig:fast_sols']} and \ref{['fig:slow_sols']} approximates the true FE solution to a very high accuracy.
• Figure 4: The relative errors in energy norm as a function of the singular value cut-off index $\sigma_r$ for the geometry of Fig. \ref{['fig:geom_crown']} with $f=x_2$ and $N=3$. We tested multiple parameter vectors $y=[y_1,y_2,y_3]$ where $y_i$ were uniformly sampled from $[1,10]$. The maximum obtained errors for each $\sigma_r$ are plotted.

Theorems & Definitions (34)

• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Remark 3.1
• Remark 4.1
• Lemma 4.1: $V_\Gamma$-stability
• proof
• Lemma 4.2: Slow-fast stability
• proof