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Abstract: The time independent Schrödinger equation stems from quantum theory axioms as a partial differential equation. This work aims at providing a novel discrete geometric formulation of this equation in terms of integral variables associated with precise geometric elements of a pair of three-dimensional interlocked grids, one of them based on tetrahedra. We will deduce, in a purely geometric way, a computationally efficient discrete counterpart of the time independent Schrödinger equation in terms of a standard symmetric eigenvalue problem. Moreover boundary and interface conditions together with non homogeneity and anisotropy of the media involved are accounted for in a straightforward manner. This approach yields to a sensible computational advantage with respect to the finite element method, where a generalized eigenvalue problem has to be solved instead. Such a modeling tool can be used for analyzing a number of quantum phenomena in modern nano-structured devices, where the accounting of the real 3D geometry is a crucial issue. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem. The idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations. In this way, consistency and stability of the discrete equations are ensured. At the base, a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid, by properly choosing the dual grid. Numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field. [Copyright &y& Elsevier]

Abstract: By exploiting the geometric structure behind Maxwell’s equations, the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations, involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids: the primal grid and the dual grid. To form a finite dimensional system of equations, discrete counterparts of the constitutive relations must be introduced in addition. They are referred to as constitutive matrices which must comply with precise properties (symmetry, positive definiteness, consistency) in order to guarantee the stability and consistency of the overall finite dimensional system of equations. The aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid; these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method. [ABSTRACT FROM AUTHOR]

• We uncover hidden geometric aspects of low-order compatible numerical schemes. • Proof of the equivalence between mimetic numerical schemes and geometric approaches. • Novel geometric definition of polynomial consistency for reconstruction operators. • Proof of the existence of other dual grids besides the barycentric dual grid. • Optimization of reconstruction operators to minimize the reconstruction error. In this work, we uncover hidden geometric aspect of low-order compatible numerical schemes. First, we rewrite standard mimetic reconstruction operators defined by Stokes theorem using geometric elements of the barycentric dual grid, providing the equivalence between mimetic numerical schemes and discrete geometric approaches. Second, we introduce a novel global property of the reconstruction operators, called P 0 -consistency, which extends the standard consistency requirement of the mimetic framework. This concept characterizes the whole class of reconstruction operators that can be used to construct a global mass matrix in such a way that a global patch test is passed. Given the geometric description of the scheme, we can set up a correspondence between entries of reconstruction operators and geometric elements of a secondary grid, which is built by duality from the primary grid used in the scheme formulation. Finally, we show the that the geometric interpretation is necessary for the correct evaluation of certain physical variables in the post-processing stage. A discussion on how the geometric viewpoint allows to optimize reconstruction operators completes the exposition. [ABSTRACT FROM AUTHOR]

We present a new mimetic finite difference method for diffusion problems that converges on grids with curved (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of P 0 -consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries. • Novel mimetic finite difference method able to deal with highly curved faces. • It achieves a convergent discrete problem which is symmetric and uses one degree of freedom for each curved face. • It can deal with geometries having curved boundaries. • It introduces a novel concept of consistency applicable to mimetic finite difference methods. [ABSTRACT FROM AUTHOR]