SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers

Abstract

Considering a singularly perturbed problem with exponential and characteristic layers, we show convergence for non-standard higher-order finite elements using the streamline diffusion finite element method (SDFEM). Moreover, for the standard higher-order space Q p supercloseness of the numerical solution w.r.t. an interpolation of the exact solution in the streamline diffusion norm of order p + 1/2 is proved.

Convergence phenomena of \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements for convection–diffusion problems

“…Proof. This is a generalization of the results given in [3, 21] for the standard Shishkin‐mesh. Combining the analyses therein with techniques for S‐type meshes, see, for example, [6], yields the desired bounds.…”

“…In [3, 21], it was shown that an SDFEM solution has a supercloseness property. To be precise, the following theorem holds.…”

“…For the SDFEM and \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $\mathcal{Q}_p$ \end{document}‐elements with p > 1 applied to problem (1.1) and (1.3), supercloseness of order p + 1/2, that is where I N is a special interpolation operator, was proved in [21]. In [3], the author showed a similar analysis to succeed for (1.1) and (1.2) too. In both papers, stabilization was needed to prove the result.…”

Local projection stabilized and characteristic decoupled scheme for the fluid–fluid interaction problems

“…For time‐dependent problems, we use the SD method discrete only in space variables and the finite difference discrete in time direction to derive the finite difference streamline diffusion (FDSD) method . This method keeps the essential aspect of the original SD method and simplifies the algorithm structure . However, the SD/FDSD methods have some undesirable features: introducing additional nonphysical coupling terms between velocity and pressure; producing inaccurate numerical solutions near the boundary; having to calculate second derivative when using high order elements.…”

Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems

“…For higher order methods using Q p -elements with p > 1 so far only for the streamline diffusion method supercloseness results are known. In [3] it was proven that π N p u − u N ε ≤ C(N −1 ln N) p+1/2 ln N holds for the streamline diffusion solution u N in the case of Q p -elements on a suitable piecewise uniform Shishkin mesh and conditions on the stabilisation parameters. The interpolant π N p is a so called vertex-edge-cell interpolant [9,15].…”