Executive Summary : | Electromagnetic phenomena play a crucial role in modern technology, making the design and analysis of schemes for the approximate solution of electromagnetic field problems a core discipline of numerical mathematics and scientific computing. Maxwell's equations, which are fundamental governing equations of electromagnetism, are encountered in the form of coupled magnetic and electric equations. Finite element discretizations of these equations are not simple, as they are highly sensitive to the conformity of approximation spaces. Over the last two decades, many alternative approaches have been developed, including H(curl) conforming edge element methods, H1 conforming nodal finite element methods with weighted regularization, the singular complement/field method, interior penalty methods, nonconforming finite element methods, and mixed finite element methods. However, in most real applications, Maxwell's equations are imposed in heterogonous media, leading to jump discontinuities across interfaces in the domain of interest, leading to a solution not in H1.
Of the existing finite element methods, most are not directly applicable or become inefficient for the interface problem due to low global regularity of the true solution and irregular shape of the interfaces. This project proposes developing higher order weak Galerkin finite element schemes for the time-dependent Maxwell interface problems, accommodating very irregular interfaces and low regular solutions. |