In this work, we develop a numerical scheme for a class of singularly perturbed quasilinear problems with integral boundary conditions. The quasilinear equation is discretized using a hybrid scheme, and the composite trapezoidal rule is used to discretize the boundary condition. We construct a general error analysis framework for the discrete scheme. Within this framework, the discrete scheme is shown to be uniformly convergent of $\mathcal{O}(N^{-2}{\ln }^{2}N)$ on Shishkin meshes and $\mathcal{O}(N^{-2})$ on Bakhvalov meshes. Further, we propose adaptive generation of meshes based on a suitable monitor function and the mesh equidistribution principle. We prove that on these meshes the discrete scheme is uniformly convergent of $\mathcal{O}(N^{-2}).$ Our theoretical findings are supported by numerical results obtained through experiments.

中文翻译：

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具有积分边界条件的奇异摄动拟线性问题的高阶收敛方法

在这项工作中，我们开发了一类具有积分边界条件的奇异摄动拟线性问题的数值方案。拟线性方程采用混合格式离散化，边界条件采用复合梯形法则离散化。我们为离散方案构建了一个通用的误差分析框架。在这个框架内，离散方案被证明是一致收敛的$\mathcal{○}({ñ}^{-2}{\ln }^2ñ)$在 Shishkin 网格和$\mathcal{○}({ñ}^{-2})$在 Bakhvalov 网格上。此外，我们提出了基于合适的监控功能和网格均匀分布原理的网格自适应生成。我们证明了在这些网格上离散方案是一致收敛的$\mathcal{○}({ñ}^{-2}).$我们的理论发现得到了通过实验获得的数值结果的支持。