Abstract
It is well known that the standard discretization methods usually suffer from several numerical instabilities when simulating advection-dominated flows, requiring some enhancements. Therefore, stabilized formulations and/or adaptive mesh strategies are needed to suppress nonphysical oscillations. Beyond the stabilized formulations, additional special techniques, e.g., shock-capturing operators, are also required for resolving sharp layers accurately. For these reasons, asymptotic methods can be preferred as an alternative to numerical schemes for simulating such phenomena, especially when the behavior of solutions within and outside the boundary/inner layers is of interest. To that end, in this computational work, we adopt an asymptotic approach called the successive complementary expansion method (SCEM) and are interested in boundary layer solutions of elliptic partial differential equations. The regularly perturbed boundary-value problems arising from the SCEM process are discretized with the classical Galerkin finite element method. In this regard, we propose an asymptotic-numerical hybrid scheme. The results obtained reveal that the proposed method works quite well for moderately small values of the diffusion (perturbation) parameter. Besides, it achieves that even at the zeroth-order approximation. On the other hand, it is also observed that, depending on the performance of the complementary numerical scheme, the hybridized method needs additional treatment for challenging values of the diffusion parameter. Therefore, we also discuss possible remedies in detail.
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Cengizci, S., Natesan, S. Hybridized successive complementary expansions for solving convection-dominated 2D elliptic PDEs with boundary layers. Comp. Appl. Math. 42, 273 (2023). https://doi.org/10.1007/s40314-023-02411-w
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DOI: https://doi.org/10.1007/s40314-023-02411-w