A robust multiscale finite element method is proposed for singularly perturbed convection–reaction–diffusion problems, featuring high accuracy, computational efficiency, and numerically observed parameter-uniform superconvergence. Within the multiscale finite element framework, multiscale basis functions are enriched with adequate local fine-scale information to enable global model reduction. The nontrivial relation between standard finite element basis functions and multiscale basis functions is analyzed and encoded in a global mapping matrix for practical implementation. By utilizing parameter-independent layer-adapted graded meshes, large errors induced by boundary layers are effectively suppressed. Numerical comparisons against existing approaches confirm the efficiency, reliability, and applicability of the presented scheme. Further numerical tests demonstrate this parameter-uniform superconvergence on graded meshes, as well as its promising capability for simulations in complex media and high-dimensional singularly perturbed systems.
Citation: Meiling Sun, Jin Zhang, Guoyun Zhang, Shan Jiang. Parameter-uniform superconvergence of multiscale simulation on layer-adapted meshes for singularly perturbed problems[J]. Networks and Heterogeneous Media, 2026, 21(6): 2286-2309. doi: 10.3934/nhm.2026085
A robust multiscale finite element method is proposed for singularly perturbed convection–reaction–diffusion problems, featuring high accuracy, computational efficiency, and numerically observed parameter-uniform superconvergence. Within the multiscale finite element framework, multiscale basis functions are enriched with adequate local fine-scale information to enable global model reduction. The nontrivial relation between standard finite element basis functions and multiscale basis functions is analyzed and encoded in a global mapping matrix for practical implementation. By utilizing parameter-independent layer-adapted graded meshes, large errors induced by boundary layers are effectively suppressed. Numerical comparisons against existing approaches confirm the efficiency, reliability, and applicability of the presented scheme. Further numerical tests demonstrate this parameter-uniform superconvergence on graded meshes, as well as its promising capability for simulations in complex media and high-dimensional singularly perturbed systems.
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Figures(13) / Tables(4)
Meiling Sun, Jin Zhang, Guoyun Zhang, Shan Jiang. Parameter-uniform superconvergence of multiscale simulation on layer-adapted meshes for singularly perturbed problems[J]. Networks and Heterogeneous Media, 2026, 21(6): 2286-2309. doi: 10.3934/nhm.2026085
Domain decomposition and the associated solutions
Global nodes and local nodes, with $ N = 2 $ and $ M = 2 $
Exact solution, FEM(S) solution on the fine mesh ($ N^2M^2 = 256^2 $), and MsFEM(S) solution on the coarse mesh ($ N^2 = 64^2 $) (Example 1, $ \varepsilon = 10^{-4} $)
FEM(G) solution on the fine mesh ($ N^2M^2 = 192^2 $) and MsFEM(G) solution on the coarse mesh ($ N^2 = 48^2 $) (Example 1, $ \varepsilon = 10^{-4} $)
Pointwise errors of FEM(S) solution on the fine mesh ($ N^2M^2 = 256^2 $) and MsFEM(S) solution on the coarse mesh ($ N^2 = 64^2 $) (Example 1, $ \varepsilon = 10^{-4} $)
Pointwise errors of FEM(G) solution on the fine mesh ($ N^2M^2 = 192^2 $) and MsFEM(G) solution on the coarse mesh ($ N^2 = 48^2 $) (Example 1, $ \varepsilon = 10^{-4} $)
$ L^\infty $ and $ H^1 $ norms of FEM(G) error and MsFEM(G) error on a log-log scale (Example 1, $ \varepsilon = 10^{-4} $)
Exact solution, FEM(G) solution on the fine mesh ($ N^2M^2 = 644^2 $), and MsFEM(G) solution on the coarse mesh ($ N^2 = 161^2 $). (Example 2, $ \varepsilon = 10^{-3} $, top: 3D mesh figure, bottom: 2D surf contour)
Pointwise errors of FEM(G) solution on the fine mesh ($ N^2M^2 = 644^2 $) and MsFEM(G) solution on the coarse mesh ($ N^2 = 161^2 $). (Example 2, $ \varepsilon = 10^{-3} $, top: 3D mesh figure, bottom: 2D surf contour)
$ L^2 $ and energy norms of FEM(G) error and MsFEM(G) error on a log-log scale (Example 2, $ \varepsilon = 10^{-3} $)
Exact solution, FEM(G) solution on the fine mesh ($ N^2M^2 = 280^2 $), and MsFEM(G) solution on the coarse mesh ($ N^2 = 70^2 $). (Example 2, $ \varepsilon = 10^{-6} $, top: 3D mesh figure, bottom: 2D surf contour)
Pointwise errors of FEM(G) solution on the fine mesh ($ N^2M^2 = 280^2 $) and MsFEM(G) solution on the coarse mesh ($ N^2 = 70^2 $). (Example 2, $ \varepsilon = 10^{-6} $, top: 3D mesh figure, bottom: 2D surf contour)
CPU time of FEM(G) and MsFEM(G) for Example 2, $ \varepsilon = 10^{-3} $
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