Supercloseness of linear streamline diffusion finite element method on Bakhvalov-type mesh for singularly perturbed convection-diffusion equation in 1D

Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

“…Besides, through (12), Hölder inequalities, and inverse inequalities and recalling ( 22) and ( 23), one has…”

“…In engineering practice, singularly perturbed problems have a wide range of applications [1–16]. For these problems, there usually exist one or more thin regions in their exact solutions, where the solutions vary dramatically.…”

“…holds. It is worth mentioning that the exact solution u of (1) typically features an interior layer of width (𝜀 ln(1∕𝜀)) near x = d. In engineering practice, singularly perturbed problems have a wide range of applications [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. For these problems, there usually exist one or more thin regions in their exact solutions, where the solutions vary dramatically.…”

“…With the in-depth development of singular perturbation problems, there are many stabilization methods, the most typical of which is streamline diffusion finite element methods (SDFEMs) [10,12,[19][20][21][22][23][24]. Combining SDFEMs and layer adapted meshes, we can obtain a numerical solution with satisfactory stability and accuracy; see, for example, Liu and Zhang [22] and Babu and Ramanujam [25].…”