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

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 $h$- and $hp$-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 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

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

A postprocessing technique for a discontinuous Galerkin discretization of time-dependent Maxwell's equations

We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or space. The key ideas behind this postprocessing technique stem from hybridizable discontinuous Galerkin (HDG) methods, which are equivalent to the analyzed DG scheme for specific choices of penalization parameters. We present several numerical experiments that highlight the superconvergence properties of the postprocessed electromagnetic field approximation

A controllability method for Maxwell's equations

We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached

A Painless Automatic hp-Adaptive Strategy for Elliptic Problems

In this work, we introduce a novel hp-adaptive strategy. The main goal is to minimize the complexity and implementational efforts hence increasing the robustness of the algorithm while keeping close to optimal numerical results. We employ a multi-level hierarchical data structure imposing Dirichlet nodes to manage the so-called hanging nodes. The hp-adaptive strategy is based on performing quasi-optimal unre finements. Taking advantage of the hierarchical structure of the basis functions both in terms of the element size h and the polynomial order of approximation p, we mark those with the lowest contributions to the energy of the solution and remove them. This straightforward unrefi nement strategy does not need from a fi ne grid or complex data structures, making the algorithm flexible to many practical situations and existing implementations. On the other side, we also identify some limitations of the proposed strategy, namely: (a) data structures only support isotropic h-re nements (although p-anisotropic re nements are enabled), (b) we assume certain quasi-orthogonality properties of the basis functions in the energy norm, and (c) in this work, we restrict to symmetric and positive defi nite problems. We illustrate these and other advantages and limitations of the proposed hp-adaptive strategy with several one-, two- and three-dimensional Poisson examples

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

High-frequency behaviour of corner singularities in Helmholtz problems

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory