Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equations with Inhomogeneous Boundary Conditions

TL;DR

The authors address multiscale convection-diffusion problems with inhomogeneous boundary conditions and high-contrast coefficients by developing a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). The method builds an auxiliary space and a multiscale space via energy minimization, and introduces boundary correctors to handle inhomogeneous Dirichlet, Neumann, and Robin data, with an analysis showing first-order convergence in the energy norm and second-order convergence in the $L^2$ norm; for time-dependent problems, boundary correctors are updated in time, and backward Euler schemes (CD- and D-approaches) are compared. The paper also extends the framework to nonlinear problems using Strang splitting, demonstrating exponential decay of correctors and robust performance across inflow/outflow scenarios. Overall, the work provides a rigorous, efficient multiscale solver for convection-diffusion with complex boundary conditions, applicable to both linear and nonlinear, time-dependent problems, with strong numerical validation of the theoretical rates.

Abstract

In this paper, we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for convection-diffusion equations with inhomogeneous Dirichlet, Neumann and Robin boundary conditions, along with high-contrast coefficients. For time independent problems, boundary correctors $\mathcal{D}^m$ and $\mathcal{N}^{m}$ for Dirichlet, Neumann, and Robin conditions are designed. For time dependent problems, a scheme to update the boundary correctors is formulated. Error analysis in both cases is given to show the first-order convergence in energy norm with respect to the coarse mesh size $H$ and second-order convergence in $L^2-$norm, as verified by numerical examples, with which different finite difference schemes are compared for temporal discretization. Nonlinear problems are also demonstrated in combination with Strang splitting.

Figures (6)

• Figure 1: Domain $K_i$ and oversampled domain $K^+_i$
• Figure 2: \ref{['fig:mediumA']} Medium $\kappa$\ref{['fig:source']} The source term $f$
• Figure 3: (\ref{['fig:Neum_bound']}) Example 2 (\ref{['fig:Neum_bound_inflow']}) Example 3 (\ref{['fig:Neum_bound_outflow']}) Example 4
• Figure 4: Comparison of inflow and outflow case in $L^2$ norm: $l_m=3$ and $\kappa_1/\kappa_0=10^3$
• Figure 5: Comparison of inflow and outflow case in $\mathcal{A}$ norm: $l_m=3$ and $\kappa_1/\kappa_0=10^3$

Theorems & Definitions (41)

• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof