180 research outputs found
Theory and implementation of -matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels
In this work, we study the accuracy and efficiency of hierarchical matrix
(-matrix) based fast methods for solving dense linear systems
arising from the discretization of the 3D elastodynamic Green's tensors. It is
well known in the literature that standard -matrix based methods,
although very efficient tools for asymptotically smooth kernels, are not
optimal for oscillatory kernels. -matrix and directional
approaches have been proposed to overcome this problem. However the
implementation of such methods is much more involved than the standard
-matrix representation. The central questions we address are
twofold. (i) What is the frequency-range in which the -matrix
format is an efficient representation for 3D elastodynamic problems? (ii) What
can be expected of such an approach to model problems in mechanical
engineering? We show that even though the method is not optimal (in the sense
that more involved representations can lead to faster algorithms) an efficient
solver can be easily developed. The capabilities of the method are illustrated
on numerical examples using the Boundary Element Method
The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains
This paper is the second part of a threefold article, aimed at solving
numerically the Poisson problem in three-dimensional prismatic or axisymmetric
domains. In the first part of this series, the Fourier Singular Complement
Method was introduced and analysed, in prismatic domains. In this second part,
the FSCM is studied in axisymmetric domains with conical vertices, whereas, in
the third part, implementation issues, numerical tests and comparisons with
other methods are carried out. The method is based on a Fourier expansion in
the direction parallel to the reentrant edges of the domain, and on an improved
variant of the Singular Complement Method in the 2D section perpendicular to
those edges. Neither refinements near the reentrant edges or vertices of the
domain, nor cut-off functions are required in the computations to achieve an
optimal convergence order in terms of the mesh size and the number of Fourier
modes used
The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains
This is the first part of a threefold article, aimed at solving numerically
the Poisson problem in three-dimensional prismatic or axisymmetric domains. In
this first part, the Fourier Singular Complement Method is introduced and
analysed, in prismatic domains. In the second part, the FSCM is studied in
axisymmetric domains with conical vertices, whereas, in the third part,
implementation issues, numerical tests and comparisons with other methods are
carried out. The method is based on a Fourier expansion in the direction
parallel to the reentrant edges of the domain, and on an improved variant of
the Singular Complement Method in the 2D section perpendicular to those edges.
Neither refinements near the reentrant edges of the domain nor cut-off
functions are required in the computations to achieve an optimal convergence
order in terms of the mesh size and the number of Fourier modes used
On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
International audienceIn electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set of , and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale . With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived
Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient
We study the numerical approximation by edge finite elements of fields whose divergence and curl, or divergence and curl-curl, are prescribed in a bounded set of , together with a boundary condition. Special attention is paid to solutions with low-regularity, in terms of the Sobolev scale . Among others, we consider an electromagnetic-like model including an interface between a classical medium and a metamaterial. In this setting the electric permittivity, and possibly the magnetic permeability, exhibit a sign-change at the interface. With the help of T-coercivity, we address the case of a model with one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal error estimates are derived. Thanks to these results, we are also able to analyze the classical time-harmonic Maxwell equations, with one sign-changing coefficient
Système de Stokes avec flux de vitesse et pression imposés
International audienceDans cette Note, nous étudions le système de Stokes avec flux de vitesse et pression imposés, dans un domaine borné, à bord régulier par morceaux
T-coercivity: application to the discretization of Helmholtz-like problems
To solve variational indefinite problems, a celebrated tool is the Banach-NecÌŒas- BabusÌŒka theory, which relies on the inf-sup condition. Here, we choose an alternate theory, T-coercivity. This theory relies on explicit inf-sup operators, both at the continuous and discrete levels. It is applied to solve Helmholtz-like problems in acoustics and electromagnetics. We provide simple proofs to solve the exact and discrete problems, and to show convergence under fairly general assumptions. We also establish sharp estimates on the convergence rates
On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields
International audienceWe propose sharp results on the numerical approximation of low-regularity electromagnetic fields by edge finite elements. We consider general geometrical settings, including topologically non-trivial domains or domains with a non-connected boundary. In the model, the electric permittivity and magnetic per-meability are symmetric, tensor-valued, piecewise smooth coefficients. In all cases, the error can be bounded by h δ times a constant, where h is the mesh-size, for some exponent δ ∈]0, 1] that depends both on the geometry and on the coefficients. It relies either on classical estimates when δ > 1/2, or on a new combined interpolation operator when δ < 1/2. The optimality of the value of δ is discussed with respect to abstract shift theorems. In some simple configurations , typically for scalar-valued permittivity and permeability, the value of δ can be further characterized. This paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements
On the approximation of electromagnetic fields by edge finite elements. Part 3: sensitivity to coefficients
International audienceIn bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces , . Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent , such that they both belong to , for all . After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations
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