| Citation: |
Shan Jiang, Xiao Ding, Meiling Sun. PARAMETER-UNIFORM SUPERCONVERGENCE OF MULTISCALE COMPUTATION FOR SINGULAR PERTURBATION EXHIBITING TWIN BOUNDARY LAYERS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3330-3351. doi: 10.11948/20230020
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PARAMETER-UNIFORM SUPERCONVERGENCE OF MULTISCALE COMPUTATION FOR SINGULAR PERTURBATION EXHIBITING TWIN BOUNDARY LAYERS
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Abstract
We propose a multiscale finite element scheme on a graded mesh for solving a singularly perturbed convection-diffusion problem efficiently. Twin boundary layers phenomena are shown in the one-dimensional model, and an adaptively graded mesh is applied to probe the twin boundary jumps. We evoke an updated multiscale strategy through the multiscale basis functions in a linear Lagrange style. Detailed mapping behaviors are investigated on fine as well as on coarse scales, thus incorporating information at the micro-scale into the macroscopic data. High-order stability theorems in an energy norm of multiscale errors are addressed. Our approach can achieve a parameter-uniform superconvergence with limited computational costs on the coarse graded mesh. Numerical results support the high-order convergence theorem and validate the advantages over other prevalent methods in the literature, especially for the singular perturbation with very small parameters. The proposed method is twin boundary layers resolving as well as parameter uniform superconvergent.
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References
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Shan Jiang, Xiao Ding, Meiling Sun. PARAMETER-UNIFORM SUPERCONVERGENCE OF MULTISCALE COMPUTATION FOR SINGULAR PERTURBATION EXHIBITING TWIN BOUNDARY LAYERS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3330-3351. doi: 10.11948/20230020
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Figure 1.
For
$ N=4 $ $ M=2 $ $ N+1=5 $ $ N*M+1=9 $ -
Figure 2.
Exact and FEM solution on uniform mesh
$ N=40 $ $ N=30 $ $ \varepsilon=2^{-8} $ -
Figure 3.
Errors of FEM(U) on
$ N=40 $ $ N=30 $ $ \varepsilon=2^{-8} $ -
Figure 4.
Exact and FEM solution on fine graded mesh
$ N=424 $ $ N=106 $ $ \varepsilon=10^{-9} $ -
Figure 5.
Errors of FEM(G) on fine mesh
$ N=424 $ $ N=106 $ $ \varepsilon=10^{-9} $ -
Figure 6.
Convergence history for
$ Err_{\infty} $ $ Err_{en} $ $ \varepsilon=10^{-8} $ $ 10^{-10} $