Geometric and Topological Aspects of Mimetic Numerical Schemes

Mimetic or compatible numerical schemes are designed to preserve the fundamental properties of physical and mathematical models, such as conservation laws, at the discrete level. To this end, methods of algebraic topology and differential geometry play a fundamental role to design its basic building blocks, like reconstruction operators or discrete Hodge operators and their algebraic realization given by mass matrices. In this thesis we provide new geometric viewpoints of low-order compatible numerical schemes. In particular, two key principles will guide our constructions. First, a tight relation between reconstruction operators and geometric elements of the barycentric dual grid. Second, a decomposition of mass matrices as the sum of a consistent and a stabilization part. We will use these principles to extended and improve the basic building blocks at the core of mimetic numerical schemes as well as their range of applicability. We introduce the novel geometric concept of P_0-consistency which generalizes the standard consistency requirement of the mimetic methods. Fundamentally, it shows that geometric elements of a secondary grid, precisely, a barycentric dual grid, are not only useful but they are implicitly present in low-order mimetic numerical schemes, even if not made explicit. This fact has two consequences. First, it provides the equivalence between mimetic numerical schemes and discrete geometric approaches. Second, it is the key principle to extend the classical mimetic methods to grids having curved faces. Indeed, all standard mimetic methods only deal with polyhedral grids, thus having planar faces. Then, we introduce a new construction of sparse inverse mass matrices for arbitrary tetrahedral grids and possibly inhomogeneous and anisotropic materials, debunking the conventional wisdom that the barycentric dual grid prohibits a sparse representation for inverse mass matrices. In particular, we provide a unified framework for the construction of both edge and face mass matrices and their sparse inverses as the sum of consistent and a stabilization part. Next, we address the problem of computing discrete vector potentials. Currently, the most efficient methods to compute them are based on the so-called tree-cotree decomposition. However, tree-cotree techniques suffer from well-known termination problems that we show be related to topological obstructions of the three-dimensional space. We propose a new algorithm based on discrete Morse theory that is able to deal with such topological obstructions. Finally, we extend the range of applicability of mimetic numerical schemes by introducing a novel mimetic volume integral method to solve eddy current problems. Integral methods for solving eddy current problems use Biot-Savart law to produce non-local constitutive relations that lead to fully populated generalized mass matrices. Yet, these formulations are very appealing because only the mesh of conductors is needed. We show how our novel mimetic method solves the three main problems of volume integral methods. First, the computation of the inductance matrix elements is slow and also delicate because of the singularity in the integral equation. We exploit constant basis functions that allow a much faster inductance matrix construction with respect to the standard one based on the Rao-Wilton-Glisson (RWG) or Raviart-Thomas (RT) basis functions. Second, our basis functions work for polyhedral elements while producing the same results as RWG and RT basis functions for tetrahedral grids. Third, the new basis functions allow to factorize the inductance matrix and to introduce a novel family of groundbreaking low-rank inductance matrix compression techniques that show several orders of magnitude improvement in memory occupation and computational effort than state-of-the-art alternatives, allowing to solve problems that otherwise cannot be faced.