Research Projects

This proposal aims to develop and analyze finite element methods (FEMs) for singularly perturbed problems (SPPs) with discontinuous coefficients and/or boundary singularities. These problems are common in real-life phenomena, such as fluid flow problems and drift diffusion equations in electrical engineering and electron flow modeling in semiconductor devices. The exact solution of these problems exhibits boundary and/or interior layer(s), which pose difficulties for their numerical solution. Classical numerical methods based on uniform mesh give large errors and fail to capture layer behavior until the mesh size is of the order of the perturbation parameter. Over the last two decades, significant work has been done on parameter uniform finite difference schemes for the numerical solution of SPPs with discontinuous data. However, finite element schemes can be easily implemented on complex geometry, motivating the study of FEM for the numerical solution of SPPs with non-smooth data. Streamline diffusion finite element (SDFEM) and discontinuous Galerkin finite element (DGFEM) are popular stabilized finite element methods for SPPs. DGFEM allows for discontinuous piecewise polynomial for trial space approximation, while DGFEM overcomes the limitations of the Galerkin finite element method. The project proposes designing, implementing, and analyzing numerical schemes based on discontinuous Galerkin finite element methods to solve singularly perturbed problems with discontinuous coefficients.