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01.09.2018 | Ausgabe 3/2018

# Analysis of the SDFEM for singularly perturbed differential–difference equations

Zeitschrift:
Calcolo > Ausgabe 3/2018
Autoren:
Li-Bin Liu, Haitao Leng, Guangqing Long
Wichtige Hinweise
This work is supported by National Science Foundation of China (11761015, 11461011), the Natural Science Foundation of Guangxi (2017GXNSFBA198183), the key project of Guangxi Natural Science Foundation (2017GXNSFDA198014), and innovation Project of Guangxi Graduate Education (JGY2017086).

## Abstract

In this paper, the stability and accuracy of a streamline diffusion finite element method (SDFEM) for the singularly perturbed differential–difference equation of convection term with a small shift is considered. With a special choice of the stabilization quadratic bubble function and by using the discrete Green’s function, the new method is shown to have an optimal second order in the sense that $$\Vert u-u_{h}\Vert _{\infty }\le C\inf \nolimits _{v_h\in V^h}\Vert u-v_{h}\Vert _{\infty }$$, where $$u_{h}$$ is the SDFEM approximation of the exact solution u in linear finite element space $$V_{h}$$. At last, a second order uniform convergence result for the SDFEM is obtained. Numerical results are given to confirm the $$\varepsilon$$-uniform convergence rate of the nodal errors.

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