| 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 (mimic) 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. By carefully defining algebraic structures to mimic the various operations in the continuum using operators from multilinear algebra, alternative proofs to the literature are obtained for discrete versions of the integration theorems. The continuum fields are defined in a primal staggered grid, which is then linked to a dual staggered grid through a discrete Hodge star operator, which is defined as the composition of a reconstruction and a projection operator.; Using bilinear forms, appropriate inner products are defined and used to prove the integration by parts theorem, positivity of the Laplacian operator and a discrete analogue to the Friedrichs-Poincare inequality. The result is clear for regular meshes, but further investigation is necessary for non-regular meshes.; Finally, a study of the truncation error of these operators is presented, showing that the first order differential operators gradient, divergence, and curl are exact, and the second order operators depend on the truncation error of the Hodge star operator. Convergence of a boundary value problem is proved to be arbitrary order depending only on the truncation error of the discrete Hodge star operator.; In Part II, we describe the design and implementation of corner compatibility conditions for the Hagstrom-Warburton absorbing planar 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. |
