Robin Boundary Conditions for the Hodge Laplacian and Maxwell's Equations

Abstract

In this thesis we study two problems. First, we generalize the Robin boundary condition for the scalar Possoin equation to the vector case and derive two kinds of general Robin boundary value problems. We propose finite elements for these problems, and adapt the finite element exterior calculus (FEEC) framework to analyze the methods. Second, we study the time-harmonic Maxwell’s equations with impedance boundary condition. We work with the function space \mathscr{H} (curl) consisting of L 2 vector fields whose curl are square integrable in the domain and whose tangential traces are square integrable on the boundary. We will show convergence of our numerical solution using \mathscr{H} (curl)- conforming finite element methods.