Dra. Camaño-Valenzuela, Jessika

In this paper we are concerned with two topics: the formulation and analysis of the eigenvalue problem for the curlcurl operator in a multiply connected domain and its numerical approximation by means of finite elements. We prove that the curlcurl operator is self-adjoint on suitable Hilbert spaces, all of them being contained in the space for which curlvv⋅nn=0curl⁡vv⋅nn=0 on the boundary. Additional constraints must be imposed when the physical domain is not topologically trivial: we show that a viable choice is the vanishing of the line integrals of vvvv on suitable homological cycles lying on the boundary. A saddle-point variational formulation is devised and analyzed, and a finite element numerical scheme is proposed. It is proved that eigenvalues and eigenfunctions are efficiently approximated and some numerical results are presented in order to assess the performance of the method.

We construct sets of basis functions of the space of divergence-free finite elements of Raviart–Thomas type in domains of general topology. Two different methods are presented: one using a suitable selection of the curls of Nédélec finite elements, the other based on an efficient algebraic procedure. The first approach looks to be more useful for numerical approximation, as the basis functions have a localized support.

In the published article, Figure 5 corresponds to an eigenfunction associated not with the first smallest positive eigenvalue. A correct eigenfunction of the latter is depicted in Fig. 1 here. Note that this eigenfunction is axisymmetric, as can be seen from Fig. 2 where its radial, azimuthal and vertical components are plotted on different meridian sections.

In this article, we analyse an augmented mixed finite element method for the steady Navier–Stokes equations. More precisely, we extend the recent results from Camaño et al.. (2017, Analysis of an augmented mixed-FEM for the Navier–Stokes problem. Math. Comput., 86, 589–615) to the case of mixed no-slip and traction boundary conditions in different parts of the boundary, and introduce and analyse a new pseudostress–velocity-augmented mixed formulation for the fluid flow problem. The well-posedness analysis is carried out by combining the classical Babuška–Brezzi theory and Banach’s fixed-point theorem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well defined. For instance, Raviart–Thomas elements of order k≥0 k≥0 and continuous piecewise polynomials of degree k+1 k+1 for the nonlinear pseudostress tensor and velocity, respectively, yield optimal convergence rates. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the proposed discretization. The proof of reliability hinges on the global inf–sup condition and the local approximation properties of the Clément interpolant, whereas the efficiency of the estimator follows from inverse inequalities and localization via edge–bubble functions. A set of numerical results exemplifies the performance of the augmented method with mixed boundary conditions. The tests also confirm the reliability and efficiency of the estimator, and show the performance of the associated adaptive algorithm.

A new mixed variational formulation for the Navier--Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier--Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additional unknown, and the Dirichlet boundary condition. The resulting augmented scheme is then written equivalently as a fixed point equation, and hence the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. No discrete inf-sup conditions are required for the well-posedness of the Galerkin scheme, and hence arbitrary finite element subspaces of the respective continuous spaces can be utilized. In particular, given an integer K >_0, piecewise polynomials of degree _< K for the gradient of velocity, Raviart--Thomas spaces of order K for the pseudostress, and continuous piecewise polynomials of degree _< K + 1 for the velocity, constitute feasible choices. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.

The goal of this paper is to compare two computational models for the inverse problem of electroencephalography: the localization of brain activity from measurements of the electric potential on the surface of the head. The source current is modeled as a dipole whose localization and polarization has to be determined. Two methods are considered for solving the corresponding forward problems: the so called subtraction approach and direct approach. The former is based on subtracting a fundamental solution, which has the same singular character of the actual solution, and solving computationally the resulting non-singular problem. Instead, the latter consists in solving directly the problem with singular data by means of an adaptive process based on an aposteriori error estimator, which allows creating meshes appropriately refined around the singularity. A set of experimental tests for both, the forward and the inverse problems, are reported. The main conclusion of these tests is that the direct approach combined with adaptivity is preferable when the localization of the dipole is close to an interface between brain tissues with different conductivities.

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach fixed point theorem and the Lax-Milgram lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree K for the nonlinear-pseudostress tensor and continuous piecewise polynomial elements of degree K + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.

We propose and analyze an efficient algorithm for the computation of a basis of the space of divergence-free Raviart–Thomas finite elements. The algorithm is based on graph techniques. The key point is to realize that, with very natural degrees of freedom for fields in the space of Raviart–Thomas finite elements of degree r+1r+1 and for elements of the space of discontinuous piecewise polynomial functions of degree r≥0r≥0, the matrix associated with the divergence operator is the incidence matrix of a particular graph. By choosing a spanning tree of this graph, it is possible to identify an invertible square submatrix of the divergence matrix and to compute easily the moments of a field in the space of Raviart–Thomas finite elements with assigned divergence. This approach extends to finite elements of high degree the method introduced by Alotto and Perugia (Calcolo 36:233–248, 1999) for finite elements of degree one. The analyzed approach is used to construct a basis of the space of divergence-free Raviart–Thomas finite elements. The numerical tests show that the performance of the algorithm depends neither on the topology of the domain nor or the polynomial degree r.

In [F. Cakoni, D. Colton, S. Meng, and P. Monk, SIAM J. Appl. Math., 76 (2016), pp. 1737--1763] it was suggested to use Stekloff eigenvalues for the Helmholtz equation to detect changes in a scatterer using remote measurements of the scattered wave. This paper investigates the use of Stekloff eigenvalues for Maxwell's equations for the same purpose. Because the Stekloff eigenvalue problem for Maxwell's equations is not a standard eigenvalue problem for a compact operator, we propose a modified Stekloff problem that restores compactness. In order to measure the modified Stekloff eigenvalues of a domain from far field measurements we perturb the usual far field equation of the linear sampling method by using the far field pattern of an auxiliary impedance problem related to the modified Stekloff problem. We are then able to show (1) the existence of modified Stekloff eigenvalues and (2) the well-posedness of the corresponding auxiliary exterior impedance problem and (3) to provide theorems that support our claim to be able to detect modified Stekloff eigenvalues from far field measurements. Preliminary numerical results show that for some simple domains it is possible to measure a few modified Stekloff eigenvalues. (As for the Helmholtz equation, not all eigenvalues can be measured.) In addition the modified Stekloff eigenvalues are changed by perturbations of the scatterer. An open problem is to obtain a proof of the existence of modified Stekloff eigenvalues for absorbing media.

A new stress-based mixed variational formulation for the stationary Navier-Stokes equations with constant density and variable viscosity depending on the magnitude of the strain tensor, is proposed and analyzed in this work. Our approach is a natural extension of a technique applied in a recent paper by some of the authors to the same boundary value problem but with a viscosity that depends nonlinearly on the gradient of velocity instead of the strain tensor. In this case, and besides remarking that the strain-dependence for the viscosity yields a more physically relevant model, we notice that to handle this nonlinearity we now need to incorporate not only the strain itself but also the vorticity as auxiliary unknowns. Furthermore, similarly as in that previous work, and aiming to deal with a suitable space for the velocity, the variational formulation is augmented with Galerkin-type terms arising from the constitutive and equilibrium equations, the relations defining the two additional unknowns, and the Dirichlet boundary condition. In this way, and as the resulting augmented scheme can be rewritten as a fixed-point operator equation, the classical Schauder and Banach theorems together with monotone operators theory are applied to derive the well-posedness of the continuous and associated discrete schemes. In particular, we show that arbitrary finite element subspaces can be utilized for the latter, and then we derive optimal a priori error estimates along with the corresponding rates of convergence. Next, a reliable and efficient residual-based a posteriori error estimator on arbitrary polygonal and polyhedral regions is proposed. The main tools used include Raviart-Thomas and Clément interpolation operators, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions. Finally, several numerical essays illustrating the good performance of the method, confirming the reliability and efficiency of the a posteriori error estimator, and showing the desired behavior of the adaptive algorithm, are reported.