What question did this study set out to answer?

The aim is to develop a new multiscale finite element method that effectively handles convection-diffusion equations with inhomogeneous boundary conditions.

March 19, 2026Open Access

Generalized multiscale finite element methods for heterogeneous convection diffusion equations with inhomogeneous boundary conditions

Key Points

The aim is to develop a new multiscale finite element method that effectively handles convection-diffusion equations with inhomogeneous boundary conditions.
Developed a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM).
Designed boundary correctors for Dirichlet, Neumann, and Robin conditions.
Conducted error analysis for both time-independent and time-dependent problems.
Compared different finite difference schemes for temporal discretization.
Achieved first-order convergence in energy norm regarding the coarse mesh size.
Demonstrated second-order convergence in L2 norm.
Validated findings through numerical examples, including nonlinear problems with Strang splitting.

Abstract

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 Dᵐ D m and N^m 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 L²- 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.

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