Robust numerical methods for singularly perturbed parabolic PDEs with interior and boundary layers
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This thesis provides some uniformly convergent numerical methods for solving singularly perturbed convection-diffusion problems with boundary or/and interior layers. A differential equation becomes singularly perturbed when a small parameter is multiplying with the highest-order derivative. The solutions of these types of problems exhibit thin boundary or/and interior layers when the small parameter tends to zero. Because of layer appearing in the solution, the classical numerical method on the uniform mesh may fail. To construct an uniformly convergent numerical scheme to this type of problem, one has to reduce the mesh size in comparison with the small parameter.The main aim of this thesis is to apply, analyze and optimize ε-uniform fitted mesh methods (FMMs) for solving different types of singularly perturbed convection-diffusion problems with boundary or/and interior layers in 1D and 2D.
Supervisor: Natesan Srinivasan