Abstract
The motive of the present work is to propose an adaptive numerical technique for singularly perturbed convection-diffusion problem in two dimensions. It has been observed that for small singular perturbation parameter, the problem under consideration displays sharp interior or boundary layers in the solution which cannot be captured by standard numerical techniques. In the present work, Hughes stabilization strategy along with the streamline upwind/Petrov-Galerkin (SUPG) method has been proposed to capture these boundary layers. Reliable a posteriori error estimates in energy norm on anisotropic meshes have been developed for the proposed scheme. But these estimates prove to be dependent on the singular perturbation parameter. Therefore, to overcome the difficulty of oscillations in the solution, an efficient adaptive mesh refinement algorithm has been proposed. Numerical experiments have been performed to test the efficiency of the proposed algorithm.
Highlights
Perturbed problems occur frequently in various branches of applied science and engineering, e.g., fluid dynamics, aerodynamics, oceanography, quantum mechanics, chemical reactor theory, reaction-diffusion processes, and radiating flows
It has been observed that the streamline upwind/Petrov-Galerkin (SUPG) method provides good approximate solution in the region where there is no sharp change in the solution but fails badly in the small subregions of sharp boundary layers
Numerical experiments have been carried out in order to test the efficiency and robustness of the proposed adaptive refinement technique based on the derived error estimates
Summary
Perturbed problems occur frequently in various branches of applied science and engineering, e.g., fluid dynamics, aerodynamics, oceanography, quantum mechanics, chemical reactor theory, reaction-diffusion processes, and radiating flows. On the basis of error estimates, the author proposed numerical solution of singularly perturbed convection-dominated problems on adaptive refined grid. Zhao et al [6] proposed adaptive numerical technique for convection-diffusion equations based on semirobust residual a posteriori error estimates for lower order nonconforming finite element approximations of streamline diffusion method. It has been observed that occurrence of these nonphysical oscillations in the region of sharp boundary layers in the discrete solution of SUPG method is based on the fact that this scheme is not monotonicity preserving To overcome this difficulty, in the present work, we have proposed Hughes stabilization strategy [8] along with the SUPG method. The a posteriori error estimates have been derived for the proposed scheme Based on these estimates, an anisotropic mesh refinement strategy has been proposed for singularly perturbed problems.
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