Numerical study of an adaptive domain decomposition algorithm based on Chebyshev tau method for solving singular perturbed problems

It is known that spectral methods offer exponential convergence for infinitely smooth solutions. However, they are not applicable for problems presenting singularities or thin layers, especially true for the ones with the location of singularity unknown. An adaptive domain decomposition method (DDM) integrated with Chebyshev tau method based on the highest derivative (CTMHD) is introduced to solve singular perturbed boundary value problems (SPBVPs). The proposed adaptive algorithm uses the refinement indicators based on Chebyshev coefficients to determine which subintervals need to be refined. Numerical experiments have been conducted to demonstrate the superior performance of the method for SPBVPs with a number of singularities including boundary layers, interior layers and dense oscillations. A fourth order nonlinear SPBVP is also concerned. The numerical results illustrate the efficiency and applicability of our adaptive algorithm to capture the locations of singularities, and the higher accuracy in comparison with some existing numerical methods in the literature. Adaptive domain decomposition method for singular perturbed problems is proposed.The proposed adaptive algorithm captures the locations of the layers effectively.Chebyshev tau method based on the highest derivative is applied.The refinement indicator is based on the tails of the Chebyshev coefficients.A numerical example with internal layers and oscillations is computed.