Abstract: This work deals with the characterization and estimation of the state of charge (SoC) in a lithium-ion battery. A hysteresis model is included in the classical equivalent circuit model (ECM) for a battery system and then we improve the estimation of the state of charge by applying the Extended Kalman Filter (EKF). The hysteretic behavior is modelled with the classical Preisach model used for magnetic materials. The construction of the Preisach operator is made by means of the Everett function identified from experimental data which only involve the charging curves of the battery. Thus, a significant reduction in the time necessary to obtain the measurements is achieved. The model is assessed with some laboratory experiments performed on a lithium-ion battery and the results show that with this procedure hysteresis is well reproduced, even when interior loops are present. In addition, the use of the EKF allows us to eliminate the measurements noise and ensure the accuracy of SoC estimation. The high computational efficiency and precision of the method, joined to the limited computational resources needed for the numerical implementation, make it particularly suitable for real-time embedded battery management system (BMS) applications.

Abstract: In this talk, we develop a virtual element discretization to approximate the steady quasi-geostrophic equations of the ocean in stream-function form. We write a variational formulation and propose a C¹-conforming virtual discretization. We prove that the discrete problem is well-posed by using the Banach fixed-point Theorem and assuming smallness of the data. Under standard assumptions on the computational domain, we establish error estimates in H²-norm for the stream-function. Finally, we include some numerical experiments that illustrate the behavior of the virtual scheme.

Abstract: We analyze, on two dimensional polygonal domains, classical low--order inf-sup stable finite element approximations of the stationary Navier-Stokes equations with singular sources. We operate under the assumption that the continuous and discrete solutions are sufficiently small. On convex domains, we perform an a priori error analysis. On Lipschitz, but not necessarily convex, polygonal domains, we design an a posteriori error estimator and prove that is globally reliable and locally efficient. We illustrate the theory with numerical tests.

Abstract: Se consideran leyes de conservación escalares modelando dinámica de tráfico vehicular o sedimentación de partículas en un espesador-clarificador unidimensional. La existencia de las soluciones es establecida probando la convergencia de un esquema de volúmenes finitos conservativos para ambos casos. Simulaciones numéricas son presentadas.

Abstract: A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure, respectively. The a priori error analysis is carried out in the L²-norm for vorticity, pressure, and velocity; under a smallness assumption either on the convecting velocity, or on the mesh parameter. Finally, a set of numerical examples in 2D and 3D is given.

Abstract: We propose and analyze a new mixed formulation for the Brinkman--Forchheimer equations for unsteady flows. Besides the velocity, our approach introduces the velocity gradient and pseudostress tensors as further unknowns. As a consequence, we obtain a three-field mixed variational formulation presenting a Banach spaces framework. We establish existence and uniqueness of a solution to the weak formulation, together with the corresponding stability bounds, employing classical results on nonlinear monotone operators. We then present well-posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart--Thomas spaces of degree k ≥0 for the pseudostress tensor and discontinuous piecewise polynomials of degree k for the velocity and velocity gradient tensor, and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of domain configurations and model parameters.

Abstract: En este trabajo estudiamos un problema de convección natural de doble difusión en medios porosos dado por un sistema de tipo Navier-Stokes/Darcy, para describir la velocidad y la presión, acoplado a una ecuación vectorial de advección-difusión que describe el calor y la concentración de una cierta sustancia, de un fluido viscoso en un medio poroso con condiciones de frontera físicas. El problema del modelo se reescribe en términos de un sistema de primer orden, sin la presión, basado en la introducción del tensor de deformación y un tensor de pseudoesfuerzo no lineal en las ecuaciones de fluidos. Después de un enfoque variacional, el modelo débil resultante se aumenta utilizando términos de penalización redundantes apropiados para las ecuaciones del fluido junto con una formulación primal estándar para la concentración de sustancia y el calor. Luego, se reescribe como un problema de punto fijo equivalente. Se establecen resultados de buen planteamiento tanto para el esquema continuo como para el discreto, así como la convergencia respectiva bajo ciertos supuestos de regularidad combinados con el teorema de Lax-Milgram y los teoremas del punto fijo de Banach y Brouwer. En particular, los elementos finitos de Raviart-Thomas de orden k son utilizados para aproximar el tensor de pseudo-esfuerzos, polinomios a trozos de grado ≤ k y ≤ k+1 se utilizan para aproximar el tensor de deformación y la velocidad, respectivamente, mientras que el calor y la concentración de sustancias se aproximan mediante elementos finitos de Lagrange de orden ≤ k+1. Se derivan teóricamente estimaciones de error a priori óptimas y se confirman experimentalmente a través de algunos ejemplos numéricos que permiten además ilustrar el rendimiento del esquema mixto-primal semi-aumentado propuesto.

Abstract: We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the traffic flow over the on-ramp and off-ramps. We approximate the problem using a upwind type numerical scheme, we provide L^∞ and BV estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.

Abstract: The aim of this talk is to analyze a C¹ Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.

Abstract: In this work we propose and analyze a nonconforming mixed finite element method for the numerical simulation of the Boussinesq problem, describing the motion of a non-isothermal incompressible fluid subject to a heat source. We consider the standard velocity-pressure formulation for the fluid flow equation and the dual-mixed one for the heat equation. In this way, the unknowns of the resulting formulation are given by the velocity, the pressure, the temperature and the gradient of the latter. The corresponding Galerkin method is based on using Raviart-Thomas elements of order k≥1 for the velocity and the temperature gradient, and discontinuous elements of order k for the pressure and temperature. The H¹-conformity of the velocity is enforced by a discontinuous Galerkin approach. The resulting numerical scheme yields exactly divergence free velocity approximations; thus, it is probably energy-stable without the need to modify the underlying differential equations. To prove the unique solvability of the continuous and discrete problems we introduce an equivalent fixed-point setting and apply the classical Banach fixed-point theorem, combined with the Babuška-Brezzi theory. Finally, we derive optimal error estimates in the mesh size for smooth solutions and provide several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence.

Abstract: We design a virtual element method for the numerical approximation of the two dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element is based on the lowest-order virtual element space which contains the subspace of linear polynomials. The connection between nonnegativity between degrees of freedom and the nonnegativity of the virtual element functions is established by Maximum and Minimum Principle theorem. We prove the well-posedness of the resulting scheme using minimization of quadratic functionals. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time variables on three different mesh families.

Abstract: This talk deals with the analysis of an augmented mixed finite element method in terms of velocity-vorticity-pressure for the Navier-Stokes equations. The weak formulation is based on the introduction of suitable least squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show the wellposedness of the continuous and discrete formulations. In addition, a priori error estimates and the corresponding rates of convergence are given. Finally, we report a set of numerical examples illustrating the behaviour of the proposed scheme.

Abstract: In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart--Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme.

Abstract: The aim of this paper is to study the numerical approximation of the displacement formulation of the acoustic eigenvalue problem in the axisymmetric case. We show that spurious eigenvalues appears when lowest order triangular Raviart-Thomas elements are used to discretize the problem. We propose an alternative weak formulation of the spectral problem which allows us to avoid this drawback. A discretization based on the same finite elements is proposed and analyzed. Quasi-optimal order spectral convergence is proved, as well as absence of spurious modes. Numerical experiments are reported which agree with the theoretical results.

Abstract: El conocido modelo de flujo de tráfico Lighthill-Whitham-Richards (LWR) modela la evolución de la densidad local de automóviles mediante una ley de conservación escalar no lineal. La transición entre regímenes de flujo libre y congestionado puede describirse mediante una función de flujo o velocidad que tiene una discontinuidad a una densidad determinada. Se construye un esquema descomponiendo la función de velocidad discontinua en una función continua de Lipschitz más una función de Heaviside y diseñando un esquema de splitting correspondiente. La parte del esquema relacionada con el flujo discontinuo se maneja mediante un paso semi-implícito que, sin embargo, no involucra la solución de sistemas de ecuaciones lineales o no lineales.