Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been perfomed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.Comment: 21 pages, 10 tables, 4 figure

A two-level enriched finite element method for a mixed problem

The simplest pair of spaces is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is symmetric, locally mass conservative and keeps the degrees of freedom of the original interpolation spaces. First, we assume local enrichments exactly computed and we prove uniqueness and optimal error estimates in natural norms. Then, a low cost two-level finite element method is proposed to effectively obtain enhancing basis functions. The approach lays on a two-scale numerical analysis and shows that well-posedness and optimality is kept, despite the second level numerical approximation. Several numerical experiments validate the theoretical results and compares (favourably in some cases) our results with the classical Raviart-Thomas elemen

A virtual element method for the solution of 2D time-harmonic elastic wave equations via scalar potentials

In this paper, we propose and analyse a numerical method to solve 2D Dirichlet timeharmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure (P-) and Shear (S-) waves via a suitable Helmholtz– Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the L 2 -norm for the displacement field, in which the contributions to the error associated with the P- and S- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the P- and S- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented

Sampling methods for low-frequency electromagnetic imaging

For the detection of hidden objects by low-frequency electromagnetic imaging the Linear Sampling Method works remarkably well despite the fact that the rigorous mathematical justification is still incomplete. In this work, we give an explanation for this good performance by showing that in the low-frequency limit the measurement operator fulfills the assumptions for the fully justified variant of the Linear Sampling Method, the so-called Factorization Method. We also show how the method has to be modified in the physically relevant case of electromagnetic imaging with divergence-free currents. We present numerical results to illustrate our findings, and to show that similar performance can be expected for the case of conducting objects and layered backgrounds

Application of the scalar and vector potentials to the aerodynamics of jets

The applicability of a method based on the Stokes potentials (vector and scalar potentials) to computations associated with the aerodynamics of jets was examined. The aerodynamic field near the nozzle could be represented and that the influence of a nonuniform velocity profile at the nozzle exit plane could be determined. Also computations were made for an axisymmetric jet exhausting into a quiescient atmosphere. The velocity at the axis of the jet, and the location of the half-velocity points along the jet yield accurate aerodynamic field computations. Inconsistencies among the different theoretical characterizations of jet flowfields are shown

A unifying perspective: the relaxed linear micromorphic continuum

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict non-polar size-effects. It is intended for the homogenized description of highly heterogeneous, but non polar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models