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# Dra. Camaño-Valenzuela, Jessika

Nombre de publicación

Dra. Camaño-Valenzuela, Jessika

Nombre completo

Camaño Valenzuela, Jessika Pamela

Facultad

Email

jecamano@ucsc.cl

ORCID

18 results

## Research Outputs

Now showing 1 - 10 of 18

- PublicationAnalysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem(Numerical Methods for Partial Differential Equations, 2021)
; ;García, CarlosOyarzúa, RicardoIn this paper, we propose and analyze a new momentum conservative mixed finite element method for the Navier–Stokes problem posed in nonstandard Banach spaces. Our approach is based on the introduction of a pseudostress tensor relating the velocity gradient with the convective term, leading to a mixed formulation where the aforementioned pseudostress tensor and the velocity are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree K for the pseudostress tensor and discontinuous piece–wise polynomial elements of degree K for the velocity. With this choice of spaces, the equilibrium equation is exactly satisfied if the external force belongs to the velocity discrete space, thus the method conserves momentum, which constitutes one of the main feature of our approach. For both, the continuous and discrete problems, the Banach–Nečas–Babuška and Banach's fixed-point theorems are employed to prove unique solvability. We also provide the convergence analysis and particularly prove that the error decay with optimal rate of convergence. 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. Finally, several numerical results illustrating the performance of the method are provided. - PublicationA posteriori error analysis of a momentum conservative Banach spaces based mixed-FEM for the Navier-Stokes problem(Applied Numerical Mathematics, 2022)
; ; ;Oyarzúa, RicardoVilla-Fuentes, SegundoIn this paper we develop an a posteriori error analysis of a new momentum conservative mixed finite element method recently introduced for the steady-state Navier–Stokes problem in two and three dimensions. More precisely, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces, such as local estimates, and suitable Helmholtz decompositions, we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed finite element scheme on arbitrary (convex or non-convex) polygonal and polyhedral regions. On the other hand, inverse inequalities, the localization technique based on bubble functions, among other tools, are employed to prove the efficiency of the proposed a posteriori error indicator. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported. - PublicationAn augmented mixed finite element method for the Navier-Stokes equations with variable viscosity(SIAM Journal on Numerical Analysis, 2016)
; ;Gatica, Gabriel ;Oyarzúa, RicardoTierra, GiordanoA 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. - PublicationCorrection to: Finite element approximation of the spectrum of the curl operator in a multiply connected domain(Foundations of computational mathematics, 2019)
;Alonso Rodríguez, Ana María; ;Rodríguez, Rodolfo ;Valli, AlbertoVenegas Tapia, PabloIn 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. - PublicationAssessment of two approximation methods for the inverse problem of electroencephalography(International journal of numerical analysis and modeling, 2016)
; ;Alonso-Rodríguez, A. ;Rodríguez, R.Valli, A.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. - PublicationDivergence-free finite elements for the numerical solution of a hydroelastic vibration problem(Numerical Methods for Partial Differential Equations, 2023)
;Alonso-Rodríguez, Ana; ;De Los Santos ,EduardoRodríguez , RodolfoIn this paper, we analyze a divergence-free finite element method to solve a fluid–structure interaction spectral problem in the three-dimensional case. The unknowns of the resulting formulation are the fluid and solid displacements and the fluid pressure on the interface separating both media. The resulting mixed eigenvalue problem is approximated by using appropriate basis of the divergence-free lowest order Raviart–Thomas elements for the fluid, piecewise linear elements for the solid and piecewise constant elements for the interface pressure. It is proved that eigenvalues and eigenfunctions are correctly approximated and some numerical results are reported in order to assess the performance of the method. - PublicationAn augmented stress-based mixed finite element method for the steady state Navier-Stokes equations with nonlinear viscosity(Numerical Methods for Partial Differential Equations, 2017)
; ;Gatica, Gabriel ;Oyarzúa, RicardoRuiz-Baier, RicardoA 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. - PublicationAnalysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics(Computers and Mathematics with Applications, 2022)
; ;García, CarlosOyarzúa, RicardoIn this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided. - PublicationA five-field mixed formulation for stationary magnetohydrodynamic flows in porous media(Computer Methods in Applied Mechanics and Engineering, 2023)
;Angelo, Lady; We introduce and analyze a new mixed variational formulation for a stationary magnetohydrodynamic flows in porous media problem, whose governing equations are given by the steady Brinkman–Forchheimer equations coupled with the Maxwell equations. Besides the velocity, magnetic field and a Lagrange multiplier associated to the divergence-free condition of the magnetic field, a convenient translation of the velocity gradient and the pseudostress tensor are introduced as further unknowns. As a consequence, we obtain a five-field Banach spaces based mixed variational formulation, where the aforementioned variables are the main unknowns of the system. The resulting mixed scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with classical results on nonlinear monotone operators and a sufficiently small data assumption, are applied to prove the unique solvability of the continuous and discrete systems. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. The finite element discretization involves Raviart–Thomas elements of order for the pseudostress tensor, discontinuous piecewise polynomial elements of degree for the velocity and the translation of the velocity gradient, Nédélec elements of degree for the magnetic field and Lagrange elements of degree for the associated Lagrange multiplier. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results. - PublicationAnalysis of an augmented Mixed-Fem for the Navier-Stokes problem(Mathematics of Computation, 2017)
; ;Oyarzúa, RicardoTierra, GiordanoIn 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.