Efficient a posteriori estimates for the wave equation

International audienc

Asymptotically constant-free and polynomial-degree-robust a posteriori estimates for space discretizations of the wave equation

We derive an equilibrated a posteriori error estimator for the space (semi) discretization of the scalar wave equation by finite elements. In the idealized setting where time discretization is ignored and the simulation time is large, we provide fully-guaranteed upper bounds that are asymptotically constant-free and show that the proposed estimator is efficient and polynomial-degree-robust, meaning that the efficiency constant does not deteriorate as the approximation order is increased. To the best of our knowledge, this work is the first to derive provably efficient error estimates for the wave equation. We also explain, without analysis, how the estimator is adapted to cover time discretization by an explicit time integration scheme. Numerical examples illustrate the theory and suggest that it is sharp

On high order methods for the heterogeneous Helmholtz equation

International audienceThe heterogeneous Helmholtz equation is used in Geophysics to model the propagation of a time harmonic wave through the Earth. Processing seismic data (inversion , migration...) involves many solutions of the Helmholtz equation, so that an efficient numerical algorithm is required. It turns out that obtaining numerical approximations of waves becomes very demanding at high frequencies because of the pollution effect. In the case of homogeneous media, high order methods can reduce the pollution effect significantly , enabling the approximation of high frequency waves. However, they fail to handle fine-scale heterogeneities and can not be directly applied to heterogeneous media. In this paper, we show that if the propagation medium is properly approximated using a multiscale strategy, high order methods are able to capture subcell variations of the medium. Furthermore, focusing on a one-dimensional model problem enables us to prove frequency explicit asymptotic error estimates, showing the superiority of high order methods. Numerical experiments validate our approach and comfort our theoretical results

Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

We consider interior penalty discontinuous Galerkin discretizations of timeharmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin-Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine

A simple equilibration procedure leading to polynomial-degree-robust a posteriori error estimators for the curl-curl problem

We introduce two a posteriori error estimators for Nédélec finite element discretizations of the curl-curl problem. These estimators pertain to a new Prager-Synge identity and an associated equilibration procedure. They are reliable and efficient, and the error estimates are polynomial-degree-robust. In addition, when the domain is convex, the reliability constants are fully computable. The proposed error estimators are also cheap and easy to implement, as they are computed by solving divergence-constrained minimization problems over edge patches. Numerical examples highlight our key findings, and show that both estimators are suited to drive adaptive refinement algorithms. Besides, these examples seem to indicate that guaranteed upper bounds can be achieved even in non-convex domains

Mixed finite element discretizations of acoustic Helmholtz problems with high wavenumbers

International audienceWe study the acoustic Helmholtz equation with impedance boundary conditions formulated in terms of velocity, and analyze the stability and convergence properties of lowest-order Raviart-Thomas finite element discretizations. We focus on the high-wavenumber regime, where such discretizations suffer from the so-called "pollution effect", and lack stability unless the mesh is sufficiently refined. We provide wavenumber-explicit mesh refinement conditions to ensure the well-posedness and stability of discrete scheme, as well as wavenumber-explicit error estimates. Our key result is that the condition "k^2 h is sufficiently small", where k and h respectively denote the wavenumber and the mesh size, is sufficient to ensure the stability of the scheme. We also present numerical experiments that illustrate the theory and show that the derived stability condition is actually necessary

Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small

Decay of coefficients and approximation rates in Gabor Gaussian frames

Gabor frames are a standard tool to decompose functions into a discrete sum of "coherent states", which are localised both in position and Fourier spaces. Such expansions are somehow similar to Fourier expansions, but are more subtle, as Gabor frames do not form orthonormal bases. In this work, we analyze decay properties of the coefficients of functions in these frames in terms of the regularity of the functions and their decay at infinity. These results are analogous to the standard decay properties of Fourier coefficients, and permit to show that a finite number of coherent states provide a good approximation to any smooth rapidly decaying function. Specifically, we provide explicit convergence rates in Sobolev norms, as the number of selected coherent states increases. Our results are especially useful in numerical analysis, when Gabor wavelets are employed to discretize PDE problems