Optimal uniform convergence analysis for a singularly perturbed quasilinear reaction–diffusion problem

Abstract

The standard conforming finite element methods on one type of highly nonuniform rectangular meshes are considered for solving the quasilinear singular perturbation problem -ε2(u xx + u yy ) + ƒ(x,y;u) = 0. By using a special interpolation operator and the integral identity technique, optimal uniform convergence rates of O(N –(k+1)) in the L2-norm are obtained for all k-th (k ≥ 1) order conforming tensor-product finite elements, where N is the number of intervals in both x- and y-directions. Hence Apel and Lube's suboptimal results are improved to optimal order and generalized to the quasilinear case.