Matemática

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https://hdl.handle.net/10669/280

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A basic model for the propagation of ideologies
(2022-11) Mata Boschini, Luis Diego; Salas Jiménez, Eduardo; Sánchez Peña, Fabio Ariel
Ideas and ideologies move the world and are involved in almost every aspect of human life and society. This paper presents a mathematical model for the propagation of two different ideologies in a group of people that could convert or not to one of the ideologies. This model allowed us to analyze which relations between parameters influence the survival and dominance of an ideology. The basic reproductive number was computed and numerical simulations were performed to analyze different scenarios.
A BDDC algorithm with deluxe scaling for H(curl) in two dimensions with irregular subdomains
(2015) Calvo Alpízar, Juan Gabriel
A bound is obtained for the condition number of a BDDC algorithm for problems posed in H(curl) in two dimensions, where the subdomains are only assumed to be uniform in the sense of Peter Jones. For the primal variable space, a continuity constraint for the tangential average over each interior subdomain edge is imposed. For the averaging operator, a new technique named deluxe scaling is used. Our optimal bound is independent of jumps in the coefficients across the interface between the subdomains and depends only on a few geometric parameters of the decomposition. Numerical results that verify the result are shown, including some with subdomains with fractal edges and others obtained by a mesh partitioner.
A brief survey of Higgs bundles
(2019-06) Zúñiga Rojas, Ronald Alberto
Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.
A Geometric Splitting Theorem
(2019) Rosales Ortega, José
Let G = G1...Gl be a connected noncompact semisimple Lie group with Lie algebra g = g_1+g_2+....+ g_l acting topologically transitive on a manifold M. We obtain a geometric splitting of the metric on M that consider metrics on each G_i. Also we obtained a result about the isometry group of the manifold GX~N , where ~N is the universal covering of a leaf N of the normal foliation to the G-orbits.
A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation
(2023) Sánchez Peña, Fabio Ariel; Arroyo Esquivel, Jorge; Calvo Alpízar, Juan Gabriel
We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.
A mixed-primal finite element approximation of a sedimentation–consolidation system
(2016) Álvarez Guadamuz, Mario Andrés; Gatica Pérez, Gabriel Nibaldo; Ruiz Baier, Ricardo
This paper is devoted to the mathematical and numerical analysis of a strongly cou- pled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection — nonlinear diffusion equation describing the transport of the solids volume fraction. The variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning vari- ational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart–Thomas approximations of order k for the stress andpiecewise continuous polynomials of order k + 1 for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang- type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed- primal finite element method, and the properties of the model.
A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport.
(2021-01) Álvarez Guadamuz, Mario Andrés; Gatica Pérez, Gabriel Nibaldo; Ruiz Baier, Ricardo
This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy’s law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixedprimal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.
A Multilayer Network Model implementation for COVID-19
(2021) Calvo Alpízar, Juan Gabriel; Sánchez Peña, Fabio Ariel; Barboza Chinchilla, Luis Alberto; García Puerta, Yury Elena; Vásquez Brenes, Paola Andrea
We present a numerical implementation for a multilayer network to model the transmission of Covid-19 or other diseases with a similar transmission mechanism. The model incorporates different contact types between individuals (household, social and sporadic networks) and includes an SEIR-type model for the transmission of the virus. The algorithm described in this paper includes the main ideas of the model used to give public health authorities an additional tool for the decision-making process in Costa Rica by simulating extensive possible scenarios and projections. We include two simulations: a study of the effect of restrictions on the transmission of the virus and a Costa Rica case study that was shared with the Costa Rican health authorities.
A multilayer network model of Covid-19: implications in public health policy in Costa Rica
(2022-05) Sánchez Peña, Fabio Ariel; Calvo Alpízar, Juan Gabriel; Mery Valdovinos, Gustavo Andrés; García Puerta, Yury Elena; Vásquez Brenes, Paola Andrea; Barboza Chinchilla, Luis Alberto; Pérez Rosales, María Dolores; Rivas Chaves, Tania
Successful partnerships between researchers, experts, and public health authorities have been critical to navigate the challenges of the Covid-19 pandemic worldwide. In this collaboration, mathematical models have played a decisive role in informing public policy, with findings effectively translated into public health measures that have shaped the pandemic in Costa Rica. As a result of interdisciplinary and cross-institutional collaboration, we constructed a multilayer network model that incorporates a diverse contact structure for each individual. In July 2020, we used this model to test the effect of lifting restrictions on population mobility after a so-called “epidemiological fence” imposed to contain the country’s first big wave of cases. Later, in August 2020, we used it to predict the effects of an open and close strategy (the Hammer and Dance). Scenarios constructed in July 2020 showed that lifting restrictions on population mobility after less than three weeks of epidemiological fence would produce a sharp increase in cases. Results from scenarios in August 2020 indicated that the Hammer and Dance strategy would only work with 50% of the population adhering to mobility restrictions. The development, evolution, and applications of a multilayer network model of Covid-19 in Costa Rica has guided decision-makers to anticipate implementing sanitary measures and contributed to gain valuable time to increase hospital capacity.
A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains
(2019) Calvo Alpízar, Juan Gabriel
A new coarse space for a two-level overlapping Schwarz algorithm is presented for problems posed in three dimensions in the space H(curl, Ω). Previous studies for these methods are very restrictive about the geometry of the subdomains while this new space is well defined for general subdomains. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are used on the overlapping subdomains. The algorithm can be defined for any subdomain geometry and works for highly discontinuous coefficient distributions. Numerical experiments with irregular subdomains and different coefficient distributions are presented. The algorithm appears very promising even for random and discontinuous values of the coefficients.
A nonlinear relapse model with disaggregated contact rates: Analysis of a forward-backward bifurcation
(2023-09) Calvo Monge, Jimmy José; Sánchez Peña, Fabio Ariel; Calvo Alpízar, Juan Gabriel; Mena Arias, Darío Alberto
Throughout the progress of epidemic scenarios, individuals in different health classes are expected to have different average daily contact behavior. This contact heterogeneity has been studied in recent adaptive models and allows us to capture the inherent differences across health statuses better. Diseases with reinfection bring out more complex scenarios and offer an important application to consider contact disaggregation. Therefore, we developed a nonlinear differential equation model to explore the dynamics of relapse phenomena and contact differences across health statuses. Our incidence rate function is formulated, taking inspiration from recent adaptive algorithms. It incorporates contact behavior for individuals in each health class. We use constant contact rates at each health status for our analytical results and prove conditions for different forward-backward bifurcation scenarios. The relationship between the different contact rates heavily influences these conditions. Numerical examples highlight the effect of temporarily recovered individuals and initial conditions on infected population persistence.
A nonperturbative form of the spectral action principle in noncommutative geometry
(1998-07) Figueroa González, Héctor; Gracia Bondía, José M.; Lizzi, Fedele; Várilly Boyle, Joseph C.
Using the formalism of superconnections, we show the existence of a bosonic action functional for the standard K-cycle in noncommutative geometry, giving rise, through the spectral action principle, only to the Einstein gravity and Standard Model Yang-Mills-Higgs terms.
A numerical implementation for the high-order 2D Virtual Element Method in MATLAB
(2021) Herrera Garro, César; Corrales Barquero, Ricardo; Arroyo Esquivel, Jorge; Calvo Alpízar, Juan Gabriel
We present a numerical implementation for the Virtual Element Method that in- corporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic func- tions is discussed, based on high-order projections. This approach significantly improves running times compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations.
A partial differential equation model with age-structure and nonlinear recidivism: Conditions for a backward bifurcation and a general numerical implementation
(2019-12-15) Sánchez Peña, Fabio Ariel; Calvo Alpízar, Juan Gabriel; Segura Ugalde, Esteban; Feng, Zhilan
We formulate an age-structured three-staged nonlinear partial differential equation model that features nonlinear recidivism to the infected (infectious) class from the temporarily recovered class. Equilibria are computed, as well as local and global stability of the infection-free equilibrium. As a result, a backward-bifurcation exists under necessary and sufficient conditions. A generalized numerical framework is established and numerical experiments are explored for two positive solutions to exist in the infectious class.
A posteriori error analysis for a viscous flow-transport problem
(2016) Álvarez Guadamuz, Mario Andrés; Gatica Pérez, Gabriel Nibaldo; Ruiz Baier, Ricardo
In this paper we develop an a posteriori error analysis for an augmented mixed-primal finite element approximation of a stationary viscous flow and transport problem. The governing system corresponds to a scalar, nonlinear convection-diffusion equation coupled with a Stokes problem with variable viscosity, and it serves as a prototype model for sedimentation-consolidation processes and other phenomena where the transport of species concentration within a viscous fluid is of interest. The solvability of the continuous mixed-primal formulation along with a priori error estimates for a finite element scheme using Raviart−Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration, have been recently established in [M. Alvarez et al., ESAIM: M2AN 49 (2015) 1399–1427]. Here we derive two efficient and reliable residual-based a posteriori error estimators for that scheme: for the first estimator, and under suitable assumptions on the domain, we apply a Helmholtz decomposition and exploit local approxi- mation properties of the Cl ́ement interpolant and Raviart−Thomas operator to show its reliability. On the other hand, its efficiency follows from inverse inequalities and the localization arguments based on triangle-bubble and edge-bubble functions. Secondly, an alternative error estimator is proposed, whose reliability can be proved without resorting to Helmholtz decompositions. Our theoretical results are then illustrated via some numerical examples, highlighting also the performance of the scheme and properties of the proposed error indicators
A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem
(2017-09-05) Álvarez Guadamuz, Mario Andrés; Gatica Pérez, Gabriel Nibaldo; Ruiz Baier, Ricardo
We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fullymixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68– 95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.
A posteriori error analysis of a semi-augmented finite element method for double-diffusive natural convection in porous media
(2024) Álvarez Guadamuz, Mario Andrés; Colmenares García, Eligio Antonio; Sequeira Chavarría, Filander A.
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi-augmented mixed-primal finite element method previously developed by us to numerically solve double-diffusive natural convection problem in porous media. The model combines Brinkman-Navier-Stokes equations for velocity and pressure coupled to a vector advection-diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo-stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart-Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual-based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart-Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.
A posteriori error analysis of mixed finite element methods for stress-assisted diffusion problems
(2022) Gatica Pérez, Gabriel Nibaldo; Gómez Vargas, Bryan Andrés; Ruiz Baier, Ricardo
We develop the a posteriori error analysis for mixed-primal and fully-mixed finite element methods approximating the stress-assisted diffusion of solutes in elastic materials. The systems are formulated in terms of stress, rotation and displacements for the elasticity equations, whereas the nonlinear diffusion is cast using either solute concentration (leading to a four-field mixed-primal formulation), or the triplet concentration – concentration gradient – and nonlinear diffusive flux (yielding the six-field fully-mixed variational formulation). We have addressed the well-posedness of these formulations in two recent works, also introducing discretisations based on PEERS or Arnold–Falk–Winther elements for the linear elasticity and either Lagrange, or Lagrange – Raviart-Thomas – Lagrange triplets for the approximation of the diffusion equation. Here we advocate the derivation of two efficient and reliable residual-based a posteriori error estimators focusing on the two-dimensional case. The proofs of reliability depend on adequately formulated inf–sup conditions in combination with a Helmholtz decomposition, and they also rely on the local approximation features of Clément and Raviart–Thomas interpolations. The efficiency of the estimators results from classical inverse and discrete trace inequalities together with localisation techniques based on edge- and triangle-bubble functions. The theoretical properties of these error indicators are confirmed through numerical tests, also serving to illustrate the performance of the adaptive mesh refinement.
A two level overlapping Schwarz preconditioner for discontinuous Galerkin methods
(2024) Calvo Alpízar, Juan Gabriel; Solano Córdoba, Moisés Eduardo
This article presents a two-level overlapping additive Schwarz algorithm designed to solve an elliptic problem discretized with the symmetric discontinuous Galerkin method. The algorithm allows for the use of irregular subdomains, overcoming limitations of other approaches where the coarse mesh was based on triangular elements. Additionally, a brief description of the numerical implementation of the Galerkin method is included. Numerical results validating the relevance of our algorithm are also presented, including cases where the coefficient of the differential equation is discontinuous, which is relevant in different applications.
A two-level overlapping Schwarz method for H(curl) in two dimensions with irregular subdomains
(2015-10-07) Calvo Alpízar, Juan Gabriel
A bound is obtained for the condition number of a two-level overlapping Schwarz algorithm for problems posed in H(curl) in two dimensions, where the subdomains are only assumed to be John subdomains. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are used on the overlapping subdomains. Our bound depends only on a few geometric parameters of the decomposition. This bound is independent of jumps in the coefficients across the interface between the subdomains for most of the different cases considered. Numerical experiments that verify the result are shown, including some with subdomains with fractal edges and others obtained by a mesh partitioner.

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