Parameter-uniform superconvergence of multiscale simulation on layer-adapted meshes for singularly perturbed problems

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Abstract

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.

Keywords

singular perturbation / multiscale scheme / layer-adapted mesh / uniform superconvergence

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