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In this work we propose an adaptive version of the recently introduced Mixed High-Order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism. The latter include, in particular, the numerical modeling of comb-drive and MEMS devices. Mesh adaptation is driven by newly derived, residual-based error estimators. The resulting method has several advantageous features: It supports fairly general meshes, it enables arbitrary approximation orders, and has a moderate computational cost thanks to hybridization and static condensation. The a posteriori-driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions, allowing to fully exploit the high-order of approximation. [ABSTRACT FROM AUTHOR]

This paper addresses the calculation of the resistive distribution of halo currents in three-dimensional structures of large magnetic confinement fusion machines. A Neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error. An irrotational electric field is obtained by a geometric formulation based on the electric scalar potential, whereas three geometric formulations are compared to obtain a solenoidal current density: a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed-hybrid Finite Elements. The electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre-processing it requires. To solve this challenging problem, we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state-of-the-art solutions proposed in literature. Believing that our results are useful in other fields of scientific computing, the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented. [ABSTRACT FROM AUTHOR]

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: 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 provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of 1- and 2-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations. • Novel graph-theoretic algorithm to compute discrete potentials. • Characterization of termination problems of tree-cotree techniques. • Discrete Morse theory framework applicable to general sparse matrix computations. • Linear computational complexity for meshes having known topological obstructions. [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]

• 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]