61 research outputs found
Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity
It has been extensively studied in the literature that solving Maxwell
equations is very sensitive to the mesh structure, space conformity and
solution regularity. Roughly speaking, for almost all the methods in the
literature, optimal convergence for low-regularity solutions heavily relies on
conforming spaces and highly-regular simplicial meshes. This can be a
significant limitation for many popular methods based on polytopal meshes in
the case of inhomogeneous media, as the discontinuity of electromagnetic
parameters can lead to quite low regularity of solutions near media interfaces,
and potentially worsened by geometric singularities, making many popular
methods based on broken spaces, non-conforming or polytopal meshes particularly
challenging to apply. In this article, we present a virtual element method for
solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media
with quite arbitrary polytopal meshes, and the media interface is allowed to
have geometric singularity to cause low regularity. There are two key
novelties: (i) the proposed method is theoretically guaranteed to achieve
robust optimal convergence for solutions with merely
regularity, ; (ii) the polytopal element shape can be highly
anisotropic and shrinking, and an explicit formula is established to describe
the relationship between the shape regularity and solution regularity.
Extensive numerical experiments will be given to demonstrate the effectiveness
of the proposed method
Polytopal templates for the formulation of semi-continuous vectorial finite elements of arbitrary order
The Hilbert spaces and are needed for
variational problems formulated in the context of the de Rham complex in order
to guarantee well-posedness. Consequently, the construction of conforming
subspaces is a crucial step in the formulation of viable numerical solutions.
Alternatively to the standard definition of a finite element as per Ciarlet,
given by the triplet of a domain, a polynomial space and degrees of freedom,
this work aims to introduce a novel, simple method of directly constructing
semi-continuous vectorial base functions on the reference element via polytopal
templates and an underlying -conforming polynomial subspace. The base
functions are then mapped from the reference element to the element in the
physical domain via consistent Piola transformations. The method is defined in
such a way, that the underlying -conforming subspace can be chosen
independently, thus allowing for constructions of arbitrary polynomial order.
The base functions arise by multiplication of the basis with template vectors
defined for each polytope of the reference element. We prove a unisolvent
construction of N\'ed\'elec elements of the first and second type,
Brezzi-Douglas-Marini elements, and Raviart-Thomas elements. An application for
the method is demonstrated with two examples in the relaxed micromorphic mode
An exterior calculus framework for polytopal methods
We develop in this work the first polytopal complexes of differential forms.
These complexes, inspired by the Discrete De Rham and the Virtual Element
approaches, are discrete versions of the de Rham complex of differential forms
built on meshes made of general polytopal elements. Both constructions benefit
from the high-level approach of polytopal methods, which leads, on certain
meshes, to leaner constructions than the finite element method. We establish
commutation properties between the interpolators and the discrete and
continuous exterior derivatives, prove key polynomial consistency results for
the complexes, and show that their cohomologies are isomorphic to the
cohomology of the continuous de Rham complex
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