Justo, D: High Order Mimetic Methods

In Part I, two approaches to discretize the continuumusing ideas from mimetic methods are presented. Anumerical method is called mimetic if their operatorsimitate three important results of the continuumcalculus: integration theorems, product rules, andthe existence of a double exact sequence. After areview of these properties, discrete tensor andexterior calculus are constructed based on thedefinitions of differential and integral operatorsrepresenting the respective continuum operators. Forsimplicity, only periodic functions are considered.Algebraic diagrams are introduced gradually with theproofs of mimetic properties. Alternative proofs tothe literature are obtained for discrete versions ofthe integration theorems using multilinear algebra. Astudy of the truncation error is also presented.In Part II, we describe the design and implementationof corner compatibility conditions for theHagstrom-Warburton absorbing boundary condition;applied to Maxwell's equations on polygonal domains.Numerical experiments demonstrate the robustness ofthe methods used, allowing the artificial boundary tobe arbitrarily close to the scatterer withoutcompromising convergence.