Research Projects

The study focuses on the mathematical and numerical analysis of singularly perturbed ordinary and partial differential equations. These equations exhibit multiscale character, with regions of small width exhibiting rapid changes and steep gradients. In practical applications, the first-order derivative and higher derivatives diverge at parts of the boundary due to the perturbation parameter tending to zero. Existing methods are unstable, inaccurate, and direction-dependent, but none are suitable for accurate and direction-independent discretization. The numerical solution of singular perturbation problems faces difficulties in approximation in the smooth part of the solution and boundary/interior layers. The finite difference method (FDM) and finite element method (FEM) require a mesh to sustain approximations, but locally refined meshes can be used for consistent numerical approximations. Adaptive mesh generation is another valuable tool, but requires relying on the location and width of boundary layers known in advance. The project aims to extend the study to general problems and develop higher-order adaptive schemes based on finite difference and finite element methods. The study also aims to address problems that consider the present state of the physical system and its history, as delays can occur when hidden variables and processes are not well understood but cause time-lag.