High‐order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions

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 O(N−2ln2N) on Shishkin meshes and 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 O(N−2). Our theoretical findings are supported by numerical results obtained through experiments.