2,740 research outputs found
Numerical homogenization of H(curl)-problems
If an elliptic differential operator associated with an
-problem involves rough (rapidly varying)
coefficients, then solutions to the corresponding
-problem admit typically very low regularity, which
leads to arbitrarily bad convergence rates for conventional numerical schemes.
The goal of this paper is to show that the missing regularity can be
compensated through a corrector operator. More precisely, we consider the
lowest order N\'ed\'elec finite element space and show the existence of a
linear corrector operator with four central properties: it is computable,
-stable, quasi-local and allows for a correction of
coarse finite element functions so that first-order estimates (in terms of the
coarse mesh-size) in the norm are obtained provided
the right-hand side belongs to . With these four
properties, a practical application is to construct generalized finite element
spaces which can be straightforwardly used in a Galerkin method. In particular,
this characterizes a homogenized solution and a first order corrector,
including corresponding quantitative error estimates without the requirement of
scale separation
Finite element eigenvalue enclosures for the Maxwell operator
We propose employing the extension of the Lehmann-Maehly-Goerisch method
developed by Zimmermann and Mertins, as a highly effective tool for the
pollution-free finite element computation of the eigenfrequencies of the
resonant cavity problem on a bounded region. This method gives complementary
bounds for the eigenfrequencies which are adjacent to a given real parameter.
We present a concrete numerical scheme which provides certified enclosures in a
suitable asymptotic regime. We illustrate the applicability of this scheme by
means of some numerical experiments on benchmark data using Lagrange elements
and unstructured meshes.Comment: arXiv admin note: substantial text overlap with arXiv:1306.535
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