Executive Summary : | Efficient methods are crucial in mathematics as they provide accurate approximations with minimal computational cost. However, the development and analysis of efficient methods for boundary and interior layers originated parabolic interface problems with non-smooth data are scarce. This project aims to develop and analyze efficient layer adapted methods for these challenging problems. The project will focus on decoupling discrete solution components at each time level and employing high-accuracy methods on layer adapted meshes in space for boundary and interior layers originated coupled systems of parabolic interface problems with non-smooth data. Additionally, the project will develop an efficient layer adapted method by decoupling the task of approximating the solution's components and avoiding a non-linear solver on each time level for boundary and interior layers originated coupled systems of parabolic nonlinear interface problems with non-smooth data. Theoretical results related to parameter uniform stability and accuracy will be established. The project will also study two new problems: boundary and interior layers originated two-parameter parabolic 2D convection-diffusion-reaction interface problem with non-smooth data and coupled system of parabolic 2D interface problems with non-smooth data. The project will establish the well-posedness of these problems, establish a maximum principle, study asymptotic behavior, and develop efficient layer adapted methods for the first time in the literature. |