45 research outputs found
-adaptation driven by polynomial-degree-robust a posteriori error estimates for elliptic problems
International audienceWe devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement
Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation
In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [1] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.
The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
[1] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999
Structure-preserving mesh coupling based on the Buffa-Christiansen complex
The state of the art for mesh coupling at nonconforming interfaces is
presented and reviewed. Mesh coupling is frequently applied to the modeling and
simulation of motion in electromagnetic actuators and machines. The paper
exploits Whitney elements to present the main ideas. Both interpolation- and
projection-based methods are considered. In addition to accuracy and
efficiency, we emphasize the question whether the schemes preserve the
structure of the de Rham complex, which underlies Maxwell's equations. As a new
contribution, a structure-preserving projection method is presented, in which
Lagrange multiplier spaces are chosen from the Buffa-Christiansen complex. Its
performance is compared with a straightforward interpolation based on Whitney
and de Rham maps, and with Galerkin projection.Comment: 17 pages, 7 figures. Some figures are omitted due to a restricted
copyright. Full paper to appear in Mathematics of Computatio
Computational Electromagnetism and Acoustics
It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems
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
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Schemes and estimates for the long-time numerical solution of Maxwell’s equations for Lorentz metamaterials
We consider time domain formulations of Maxwell's equations for the Lorentz model for metamaterials. The field equations are considered in two different forms which have either six or four unknown vector fields. In each case we use arguments tuned to the physical laws to derive data-stability estimates which do not require Gronwall's inequality. The resulting estimates are, in this sense, sharp. We also give fully discrete formulations for each case and extend the sharp data-stability to these. Since the physical problem is linear it follows (and we show this with examples) that this stability property is also reflected in the constants appearing in the a priori error bounds. By removing the exponential growth in time from these estimates we conclude that these schemes can be used with confidence for the long-time numerical simulation of Lorentz metamaterials.This work was supported in part by NSFC Project 11271310, NSF grant DMS-1416742, and a grant from
the Simons Foundation (#281296 to Li), in part by scheme 4 London Mathematical Society funding and in part
by the Engineering and Physical Sciences Research Council (EP/H011072/1 to Shaw)
An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains
We propose an unfitted finite element method for numerically solving the
time-harmonic Maxwell equations on a smooth domain. The model problem involves
a Lagrangian multiplier to relax the divergence constraint of the vector
unknown. The embedded boundary of the domain is allowed to cut through the
background mesh arbitrarily. The unfitted scheme is based on a mixed interior
penalty formulation, where Nitsche penalty method is applied to enforce the
boundary condition in a weak sense, and a penalty stabilization technique is
adopted based on a local direct extension operator to ensure the stability for
cut elements. We prove the inf-sup stability and obtain optimal convergence
rates under the energy norm and the norm for both the vector unknown and
the Lagrangian multiplier. Numerical examples in both two and three dimensions
are presented to illustrate the accuracy of the method