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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {D}^m\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {N}^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">N</mi> </mrow> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> 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 <i>H</i> and second-order convergence in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月17日

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  • Po Chai Wong,
  • Eric T. Chung,
  • Changqing Ye,
  • Lina Zhao

摘要

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 \(\mathcal {D}^m\) D m and \(\mathcal {N}^{m}\) 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 \(L^2-\) 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.