A second-order exponential integration constraint energy minimizing generalized multiscale method for parabolic problems

TL;DR

This work addresses semilinear parabolic PDEs in high-contrast media by coupling constraint energy minimizing generalized multiscale finite element methods (CEM-GMsFEM) for spatial discretization with explicit exponential Runge-Kutta time integration. The method constructs localized, exponentially decaying multiscale basis functions through a two-stage process: local spectral problems to form an auxiliary space and constrained energy minimization on oversampling regions. Rigorous error estimates in $H^1$ and $L^2$ norms are derived for first- and second-order exponential RK schemes, and numerical experiments confirm both spatial and temporal accuracy while enabling larger time steps than traditional implicit methods. The approach offers an efficient, stable framework for multiscale parabolic problems in high-contrast media with substantial potential for practical simulations.

Abstract

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method.

Figures (9)

• Figure 1: Illustration of the $2$D multiscale grid with a typical coarse element $K_{i}$ and oversampling domain $K_{i,2}$, the fine grid element and neighborhood $\omega_{i}$ of the node $\mathrm{x}_{i}$.
• Figure 2: Permeability fields.
• Figure 3: Relative error for the CEM-GMsFEM-EIRK1 and CEM-GMsFEM-FDBE solution with increasing the number of local multiscale basis functions (left) and the number of oversampling layers (right) for problem \ref{['eq:example1']} at final time $T=0.2$.
• Figure 4: (a) CEM-GMsFEM-EIRK1, (b) CEM-GMsFEM-FDBE, and (c) The reference solution of problem \ref{['eq:example1']} at final time $T=0.2$, using coarse grid size $H=\tfrac{1}{8}$, $4$ local multiscale basis functions and $m=4$ oversampling layers.
• Figure 5: Relative error between the CEM-GMsFEM-EIRK1 solution and the reference solution with increasing the number of local basis functions (left) and the number of oversampling layers (right) for problem \ref{['eq:example2']} at the final time $T=0.2$.

Theorems & Definitions (23)

• Definition 3.1: chung2018constraint
• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4
• Example 5.5
• Lemma A.4
• Lemma A.5
• proof
• Theorem A.6