231 research outputs found
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
time-harmonic 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
Asymptotically constant-free and polynomial-degree-robust a posteriori error estimates for time-harmonic Maxwell's equations
We propose a novel a posteriori error estimator for the N\'ed\'elec finite
element discretization of time-harmonic Maxwell's equations. After the
approximation of the electric field is computed, we propose a fully localized
algorithm to reconstruct approximations to the electric displacement and the
magnetic field, with such approximations respectively fulfilling suitable
divergence and curl constraints. These reconstructed fields are in turn used to
construct an a posteriori error estimator which is shown to be reliable and
efficient. Specifically, the estimator controls the error from above up to a
constant that tends to one as the mesh is refined and/or the polynomial degree
is increased, and from below up to constant independent of . Both bounds are
also fully-robust in the low-frequency regime. The properties of the proposed
estimator are illustrated on a set of numerical examples
Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations
We analyze the conforming approximation of the time-harmonic Maxwell's
equations using N\'ed\'elec (edge) finite elements. We prove that the
approximation is asymptotically optimal, i.e., the approximation error in the
energy norm is bounded by the best-approximation error times a constant that
tends to one as the mesh is refined and/or the polynomial degree is increased.
Moreover, under the same conditions on the mesh and/or the polynomial degree,
we establish discrete inf-sup stability with a constant that corresponds to the
continuous constant up to a factor of two at most. Our proofs apply under
minimal regularity assumptions on the exact solution, so that general domains,
material coefficients, and right-hand sides are allowed
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., ) 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 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
Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equations
We propose a new residual-based a posteriori error estimator for
discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in
first-order form. We establish that the estimator is reliable and efficient,
and the dependency of the reliability and efficiency constants on the frequency
is analyzed and discussed. The proposed estimates generalize similar results
previously obtained for the Helmholtz equation and conforming finite element
discretization of Maxwell's equations. In addition, for the discontinuous
Galerkin scheme considered here, we also show that the proposed estimator is
asymptotically constant-free for smooth solutions. We also present
two-dimensional numerical examples that highlight our key theoretical findings
and suggest that the proposed estimator is suited to drive - and
-adaptive iterative refinements.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0920
Adjoint-based formulation for computing derivatives with respect to bed boundary positions in resistivity geophysics
In inverse geophysical resistivity problems, it is common to optimize for specific resistivity values and bed boundary positions, as needed, for example, in geosteering applications. When using gradient-based inversion methods such as Gauss-Newton, we need to estimate the derivatives of the recorded measurements with respect to the inversion parameters. In this article, we describe an adjoint-based formulation for computing the derivatives of the electromagnetic fields withrespect to the bed boundary positions. The key idea to obtain this adjoint-based formulation is to separate the tangential and normal components of the field, and treat them differently. We then apply this method to a 1.5D borehole resistivity problem. We illustrate its accuracy and some of its convergence properties via numerical experimentation by comparing the results obtained with our proposed adjoint-based method vs. both the analytical results when available and a finite differences approximation of the derivative
Finite Element Simulations of Logging-While-Drilling and Extra-Deep Azimuthal Resistivity Measurements using Non-Fitting Grids
We propose a discretization technique using non-fitting grids to simulate magnetic field-based resistivity logging measurements. Non-fitting grids are convenient because they are simpler to generate and handle than fitting grids when the geometry is complex. On the other side, fitting grids have been historically preferred because they offer additional accuracy for a fixed problem size in the general case. In this work, we analyse the use of non-fitting grids to simulate the response of logging instruments that are based on magnetic field resistivity measurements using 2.5D Maxwell’s equations. We provide various examples demonstrating that, for these applications, if the finite element matrix coefficients are properly integrated, the accuracy loss due to the use of non-fitting grids is negligible compared to the case where fitting grids are employed
Finite element approximation of electromagnetic fields using nonfitting meshes for Geophysics
We analyze the use of nonfitting meshes for simulating the propagation of electromagnetic waves inside the earth with applications to borehole logging. We avoid the use of parameter homogenization and employ standard edge finite element basis functions. For our geophysical applications, we consider a 3D Maxwell’s system with piecewise constant conductivity and globally constant permittivity and permeability. The model is analyzed and discretized using both the Eand H-formulations. Our main contribution is to develop a sharp error estimate for both the electric and magnetic fields. In the presence of singularities, our estimate shows that the magnetic field approximation is converging faster than the electric field approximation. As a result, we conclude that error estimates available in the literature are sharp with respect to the electric field error but provide pessimistic convergence rates for the magnetic field in our geophysical applications. Another surprising consequence of our analysis is that nonfitting meshes deliver the same convergence rate as fitting meshes to approximate the magnetic field. Our theoretical results are numerically illustrated via 2D experiments. For the analyzed cases, the accuracy loss due to the use of nonfitting meshes islimited, even for high conductivity contrasts
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