Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Meshfree Generalized Multiscale Exponential Integration Method for Parabolic Problems

Djulustan Nikiforov, Leonardo A. Poveda, Dmitry Ammosov, Yesy Sarmiento, Juan Galvis

Abstract

This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices, which negatively affect both the computational cost and the stability of the numerical solution. We propose a novel combined approach of the meshfree Generalized Multiscale Finite Element Method (MFGMsFEM) and exponential time integration for solving such problems. MFGMsFEM provides a robust and efficient spatial approximation, allowing us to consider complex heterogeneities without constructing a coarse computational grid. At the same time, exponential integration, using the cost-effective MFGMsFEM matrix, provides a robust temporal approximation for stiff multiscale problems, allowing larger time steps. For the proposed multiscale approach, we provide a rigorous convergence analysis, including the new analysis of the MFGMsFEM spatial approximation. We conduct numerical experiments to computationally verify the proposed approach by solving linear and semi-linear flow problems in multiscale media. Numerical results demonstrate that the proposed multiscale method achieves significant reductions in computational cost and improved stability, even with larger time steps, confirming the theoretical analysis.

Meshfree Generalized Multiscale Exponential Integration Method for Parabolic Problems

Abstract

This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices, which negatively affect both the computational cost and the stability of the numerical solution. We propose a novel combined approach of the meshfree Generalized Multiscale Finite Element Method (MFGMsFEM) and exponential time integration for solving such problems. MFGMsFEM provides a robust and efficient spatial approximation, allowing us to consider complex heterogeneities without constructing a coarse computational grid. At the same time, exponential integration, using the cost-effective MFGMsFEM matrix, provides a robust temporal approximation for stiff multiscale problems, allowing larger time steps. For the proposed multiscale approach, we provide a rigorous convergence analysis, including the new analysis of the MFGMsFEM spatial approximation. We conduct numerical experiments to computationally verify the proposed approach by solving linear and semi-linear flow problems in multiscale media. Numerical results demonstrate that the proposed multiscale method achieves significant reductions in computational cost and improved stability, even with larger time steps, confirming the theoretical analysis.
Paper Structure (20 sections, 6 theorems, 105 equations, 11 figures, 8 tables)

This paper contains 20 sections, 6 theorems, 105 equations, 11 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mathematical model
  4. Fine-grid finite element approximation
  5. Temporal discretization using finite difference method
  6. Coarse meshfree generalized multiscale finite element method
  7. Meshfree coarse scale
  8. Local basis functions
  9. Global formulation
  10. The MFGMsFEM coarse interpolation
  11. Temporal discretizations combined with MFGMsFEM spatial approximation
  12. MFGMsFEM combined with finite difference (MFGMsFEM-FD)
  13. MFGMsFEM combined with exponential integrator (MFGMsFEM-EI)
  14. Convergence analysis
  15. Numerical results
  16. ...and 5 more sections

Key Result

Lemma 5.3

Suppose that the function $f$ satisfies Assumption asp:02, and the exact solution $p(\cdot,t)$ satisfies Assumption asp:01. Then, $f$ is locally-Lipschitz continuous in a strip along the exact solution $p(\cdot,t)$, i.e., for any given positive constant $C$, for any $t\in[0,T]$ and $v,w\in V$ satisfying $\max\{\|(v-p(t)\|_{H^{1}},\|w-p(t)\|_{H^{1}}\}\leq C$, where the hidden constant in eq:lem2-1

Figures (11)

  • Figure 1: Example of some coarse-scale elements $S_i$ (green) and $S_j$ (blue).
  • Figure 2: Point cloud illustration obtained with the centroidal Voronoi tessellations method. See du2002meshfree.
  • Figure 3: Illustration of high-contrast coefficient $\kappa(x)$ and multiscale functions at $\gamma = 4$. Top-left: $\kappa(x)$ in $S_i$. Top-right: shape function $W_i(x)$. Bottom-left: eigenfunction $\psi_k^{S_i}$. Bottom-right: multiscale basis function $\psi_{ik}^{\text{ms}}$.
  • Figure 4: High-contrast permeability coefficient $\kappa(x)$ for the linear case, where $\kappa_1 = 1$ is green and $\kappa_2 = 10^4$ is yellow.
  • Figure 5: Pressure distributions at the final time for the linear case. Left: reference solution. Center: MFGMsFEM-FD solution. Right: MFGMsFEM-EI solution. The multiscale approaches' parameters are $M = 5$, $N_t = 50$, and $\gamma = 3$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Lemma 5.3
  • Lemma 5.4
  • proof
  • Theorem 5.5
  • proof
  • Proposition 5.6: Properties of the semigroup henry1981geometric
  • Theorem 5.7
  • proof
  • Theorem 5.8