A Localized Orthogonal Decomposition method for heterogeneous mixed-dimensional problems

TL;DR

This work advances numerical multiscale modeling for mixed-dimensional elliptic problems with heterogeneous coefficients by adapting the Localized Orthogonal Decomposition to bulk–interface systems. It constructs problem-adapted, locally supported bases via a priori analysis and exponential decay of the prototypical basis, then localizes computations on oversampled patches to yield a practical, parallelizable method. Theoretical results show optimal coarse-mesh convergence with a localization error decaying exponentially in the patch size, and numerical experiments corroborate accuracy across smooth, oscillatory, and complex-shaped coarse meshes. The approach enables efficient, scalable simulations of fractured porous media and related multi-scale, mixed-dimensional phenomena.

Abstract

We propose a multiscale method for mixed-dimensional elliptic problems with highly heterogeneous coefficients arising, for example, in the modeling of fractured porous media. The method is based on the Localized Orthogonal Decomposition (LOD) framework and constructs locally supported, problem-adapted basis functions on a coarse mesh that does not need to resolve the coefficient oscillations. These basis functions are obtained in parallel by solving localized fine-scale problems. Our a priori error analysis shows that the method achieves optimal convergence with respect to the coarse mesh size, independent of the coefficient regularity, with an exponentially decaying localization error. Numerical experiments validate these theoretical findings and demonstrate the computational viability of the method.

Figures (8)

• Figure 2.1: An example satisfying our assumptions for $d = 2$, with subdomain segments of codimensions $c = 0, 1, 2$ indicated.
• Figure 5.1: Example of a first-order patch $\mathsf{N}_1(T^0)$, around a coarse bulk element. The shaded red area marks the bulk elements of the patch whereas the opaque red marks the element $T$. The thickened blue lines mark the interface elements belonging to the patch.
• Figure 6.1: Examples of basis functions corresponding to a coarse mesh bulk element \ref{['subfig:basis_functions_bulk']} and a coarse mesh interface element \ref{['subfig:basis_functions_interface']} using a patch spreading $\ell = 4$ layers.
• Figure 7.1: A domain with straight interfaces (blue lines) and simplicial coarse mesh elements (red thick lines) is shown in \ref{['fig:complex_shaped_domain_orig']}. A modified domain with an interface that follows the fine mesh (red thin lines) and leads to non-convex, complex-shaped coarse mesh elements is shown in \ref{['fig:complex_shaped_domain_new']}.
• Figure 8.1: The domain $\Omega$ used for the numerical experiments. Here, the blue lines are the interfaces $\Omega^1$ and the coarse mesh corresponding to $H = \tfrac{1}{4}$ is shown in red. The fine scale mesh is shown in the zoomed-in part.

Theorems & Definitions (14)

• Lemma 2.1: Properties of $a$
• proof
• Lemma 3.1: Prototypical basis
• proof : Proof of \ref{['le:protbasis']}
• Theorem 3.2: Prototypical method
• proof
• Theorem 4.1: Exponential decay
• proof
• Corollary 5.1: Local version of exponential decay
• proof