High-order Gegenbauer integral spectral element method integrated with an adaptive Chebyshev optimization strategy for solving linear singularly perturbed differential equations

We propose a high-order adaptive spectral element method based on Gegenbauer polynomial basis for solving singularly perturbed differential equations with Dirichlet boundary conditions and a boundary layer at one end. The method reduces the singularly perturbed boundary value problem into a single system of linear algebraic equations when the solution is smooth on the solution domain, or two systems of linear algebraic equations when the solution has a thin boundary layer. The core of the new adaptive strategy lies in constructing a mathematical model of the last few Gegenbauer coefficients in the Gegenbauer truncated series approximating the second-order derivative of the solution, which can be readily converted into a minimization problem in a least-squares sense and solved using fast optimization techniques. An error and convergence analysis of the method for sufficiently smooth solutions is presented. A numerical test example with various parameter settings is presented to verify the accuracy, effectiveness, and applicability of the proposed method. Numerical comparisons with other rival methods in the literature are included to demonstrate further the power of the proposed method.