High Order Mimetic Methods: and Absorbing Boundary Conditions

In Part I, two approaches to discretize the continuum using ideas from mimetic methods are presented. A numerical method is called mimetic if their operators imitate three important results of the continuum calculus: integration theorems, product rules, and the existence of a double exact sequence. After a review of these properties, discrete tensor and exterior calculus are constructed based on the definitions of differential and integral operators representing the respective continuum operators. For simplicity, only periodic functions are considered. Algebraic diagrams are introduced gradually with the proofs of mimetic properties. Alternative proofs to the literature are obtained for discrete versions of the integration theorems using multilinear algebra. A study of the truncation error is also presented. In Part II, we describe the design and implementation of corner compatibility conditions for the Hagstrom-Warburton absorbing boundary condition; applied to Maxwell's equations on polygonal domains. Numerical experiments demonstrate the robustness of the methods used, allowing the artificial boundary to be arbitrarily close to the scatterer without compromising convergence.