Meshfree multiscale method with partially explicit time discretization

TL;DR

The paper tackles efficient simulation of fractured porous media with high parameter contrast by introducing a meshfree generalized multiscale finite element method (Meshfree GMsFEM) paired with a partially explicit time discretization. It constructs a coarse multiscale space on a point-cloud, using local spectral problems and kernel-based shape functions to project the fine-grid system into a reduced coarse system, and then splits the time stepping so that nodes associated with fractures are treated implicitly while others are updated explicitly. This splitting yields a time step that is effectively independent of the fracture diffusion coefficient, while preserving accuracy as demonstrated by two tests against fine-grid references, with $L^2$ and $H^1$ errors remaining very small. The approach offers a robust, scalable framework for multiscale simulations in fractured media, balancing stability and efficiency, and is particularly suitable for meshfree implementations that control coarse-node placement. Overall, the method facilitates high-accuracy, contrast-tolerant simulations in complex fracture networks without prohibitive time-step restrictions.

Abstract

In this paper, a multiscale approach with partially explicit time discretization is proposed. The idea is to use a partially explicit time scheme, considering a filtration problem in a fractured medium, where the implicit scheme is used for nodes whose subdomains contain fractures, and the explicit scheme is used for all others. In this way, it is possible to use a time step that is independent of the diffusion coefficient for fractures. Numerical results demonstrating high accuracy of calculations are presented.

Figures (10)

• Figure 1: Computational domains $\Omega$ with the fractures $\gamma$ (red lines). Left: Test 1. Right: Test 2.
• Figure 2: Example of some coarse-scale elements $S_i$ and $S_j$.
• Figure 3: Illustration of multiscale functions. Left: shape function $W_i(x)$. Middle: eigenvector $\psi_k^{ \text{off}}$. Right: multiscale basis function $\psi_{i,k}$.
• Figure 4: Point clouds with $N = 225$, where the orange dots are nodes for $p_I$ and the blue dots are nodes for $p_E$. Left: Test 1 with $N_I = 77$ and $N_E = 148$. Right: Test 2 with $N_I = 160$ and $N_E = 65$.
• Figure 5: Solutions at the final time for Test 1. Left: reference solution. Middle: Meshfree GMsFEM with impicit method. Right: Meshfree GMsFEM with partially explicit method.