A singularly perturbed boundary-value problem is a boundary-value problem that contains a small parameter whose value cannot be approximated by setting to zero. The one-dimensional singularly perturbed quasilinear convection-diffusion problem is considered, which has a unique solution. The solution has an exponential boundary layer, which can be resolved by layer-adapted meshes, such as the Bakhvalov mesh and the Shishkin mesh.

This paper studies numerical schemes for singularly perturbed boundary-value problems on the Bakhvalov-Shishkin mesh and the Shishkin mesh. First, the authors construct a mesh-generating function, and the element sizes of the Bakhvalov-Shishkin mesh are monotonically increasing. Based on the mesh, a family of hybrid schemes is proposed. A discrete linear operator is defined and analyzed. Then, the authors show that the numerical solution from the hybrid schemes is uniform second-order convergent. In the end, numerical derivatives from the proposed numerical schemes are proved to be nearly second order.

Several examples are employed to verify error estimates and convergence rates of solutions from the class of hybrid schemes. The examples illustrate that the errors of numerical solutions and numerical derivatives are second-order convergent on Bakhvalov-Shishkin meshes. They are less than second-order convergent on Shishkin meshes, which coincide with the theoretical error estimates.

In summary, the authors propose a class of hybrid schemes for singularly perturbed quasilinear convection-diffusion problems. Their convergence and error estimates are proved and illustrated by numerical experiments. The results are interesting and helpful. However, the paper could bring more insights if the proposed methods were compared with other existing methods.