Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

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

“…For the purpose of estimating | ∫ d 0 a ′ (x)(u I − u)𝜒dx|, we use (15), (22), (23), and 𝜒(x) = ∫ x 0 𝜒 ′ (s)ds,…”

“…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.…”

“…In fact, because of the existence of these layers, traditional numerical methods are no longer suitable. To deal with this difficulty, researchers designed an effective method-layer adapted meshes, which are fine meshes inside layers and general meshes outside, see previous studies [7,15,17,18] for more details. Among them, Shishkin meshes have become popular because of their simple structure.…”