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XXV CONGRESO DE MATEMATICA CAPRICORNIO
Universidad Católica del Norte, Antofagasta, Agosto 2 - 5, 2016
Sesión Invitada de Análisis Numérico de Ecuaciones
Diferenciales Parciales
Organizada por: Gabriel N. Gatica
Centro CI2 MA, Universidad de Concepción, Chile
PROGRAMA
BLOQUE 1: Martes 2 de Agosto, de 16.15 a 18.45
16.15 - 16.45 Nelson Moraga: Stabilization of convective terms by a Darcy-Brinkman-Forch-
heimer porous model for alloy solidification and food solar drying.
16.45 - 17.15 Julio Careaga: Entropy solutions of a scalar conservation law modelling sed-
imentation in vessels with varying cross sectional area.
17.15 - 17.45 Jessika Camaño: A priori and a posteriori error analyses of a flux-based mixed-
FEM for convection-diffusion-reaction problems.
17.45 - 18.15 Elvis Gavilán: A computational approach to a spatio-temporal and gender-
structured model for hantavirus infection in rodents.
18.15 - 18.45 Gabriel N. Gatica: A posteriori error analysis of a fully-mixed formulation
for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity.
BLOQUE 2: Miércoles 3 de Agosto, de 16.15 a 18.45
16.15 - 16.45 Ricardo Oyarzúa: An augmented stress-based mixed finite element method for
the Navier-Stokes equations with variable viscosity.
16.45 - 17.15 Carlos Garcı́a: Finite element analysis of a pressure-stress formulation for
the time-domain fluid-structure interaction.
17.15 - 17.45 Luis Gatica: Analysis of a HDG method applied to n-dimensional linear Brink-
man models.
17.45 - 18.15 Eligio Colmenares: A posteriori error analysis of an augmented fully-mixed
FEM for the Boussinesq problem.
18.15 - 18.45 Juan Calvo: A Schwarz algorithm in H(curl) for irregular subdomains in 3D.
BLOQUE 3: Jueves 4 de Agosto, de 16.15 a 18.45
16.15 - 16.45 Luis M. Villada: High order numerical schemes for one-dimension non-local
conservation laws.
16.45 - 17.15 Nestor Sánchez: A priori and a posteriori error analysis of an augmented
mixed-FEM for the Navier-Stokes/Brinkman problem.
17.15 - 17.45 Felipe Vargas: A high order HDG method for Stokes flow in curved domains.
17.45 - 18.15 Vı́ctor Osores: Métodos de alto orden para sistemas hiperbólicos con productos
no conservativos, aplicados a sistemas shallow water multicapa con sedimentación
polidispersa.
18.15 - 18.45 Nelson Moraga: A new sequential algorithm for fluid mechanics and heat
transfer in complex conjugate problems solved by finite volume method.
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y de Agosto de 2016, Antofagasta, Chile
Stabilization of convective terms by a
Darcy-Brinkman-Forchheimer porous model
for alloy solidification and food solar drying
Nelson Moraga ∗
Departamento de Ingeniería Mecánica
Universidad de La Serena
Benavente 980, La Serena, Chile
Abstract
Influence of inertial and friction effects on fluid flows in porous media are accounted for by
the Darcy-Brinkman-Forchheimer (DBF) model [1-3]. The objective of this paper is to analyze
the effects of adding the correcting BF terms on the stabilization of the non-linear convective
terms in the Navier-Stokes equations. Physical examples investigated include the solution of
two phase change problems: food dehydration inside a solar dryer and binary alloy solidification in a thick walled square mold. In the first case, heat and mass diffusion of water inside a
granular food (porous media) being removed by a laminar air flow with mixed heat and mass
transfer is described by a conjugate seven PDE’s model in a solar dryer. Binary solidification of
a binary aluminum alloy with an original model of porous temperature dependent porosity and
permeability for the mushy zone is described by a five PDE’s coupled system. Finite Volume
Method allows to solve the problems with in house programs. Solutions for the fluid mechanics
are examined in terms of the evolution of stream functions in air flow and of the heat and mass
transfer by the variation in time of temperature and water content in air and in grapes inside
the solar dryer. The stabilization role of the convective terms by the temperature dependent
porosity-permeability DBF mushy zone model of solidification is examined in terms of the computing time to solve the problem against the use of the non-porous classical solidification model.
Joint work with:
David Gallardo, Departamento de Ingeníería Mecánica, Universidad de La Serena, Benavente 980,
La Serena, Chile.
Roberto Cabrales1 , Departamento de Ciencias Básicas, Universidad del Bío-Bío, Chillán, Chile.
References
[1] N. Kladis, V. Prasad, Experimental verification of Darcy-Brinkman-Forchheimer flow model
for natural convection in porous media. J. of Thermophysical and Heat Transfer, vol. 5(4), pp.
560–576, (1991).
∗ Partially
1 e-mail:
supported by CONICYT-Chile to FONDECYT 1140074 project, e-mail: [email protected]
[email protected]
1
[2] D. Dan, P. Biswal, M. Roy, T. Basak, Role of the importance of Forchheimer term for
visualization of natural convection in porous enclosures of various shapes. International Journal
of Heat Mass Transfer, vol. 97, pp. 1044–1068, (2016) .
[3] N. Moraga, G. Sánchez, J. Riquelme, Unsteady mixed convection in a vented enclosure
partially filled with two non-Darcian porous layers. Numerical Heat Transfer A, vol. 57, pp.
1–23, (2010).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
Entropy solutions of a scalar conservation law
modelling sedimentation in vessels with varying
cross sectional area
Julio Careaga∗
CI MA and Departamento de Ingeniería Matemática
Universidad de Concepción
Concepción, Chile
2
Abstract
The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be
described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation
law with a nonconvex flux function and a weight function that depends on spatial position. The
sought unknown is the local solids volume fraction. For the most important cases of vessels with
downward-decreasing cross-sectional area and flux function with at most one inflection point,
entropy solutions of this problem are constructed by the method of characteristics. Solutions
exhibit discontinuities that mostly travel at variable speed, i.e., they are curved in the spacetime plane. These trajectories are given by ordinary differential equations that arise from the
jump condition. It is shown that three qualitatively different solutions may occur in dependence
of the initial concentration. The potential application of the findings is a new method of flux
identification via settling tests in a suitably shaped vessel. Related models also arise in flows of
vehicular traffic, pedestrians, and in pipes with varying cross-sectional area. A comparison of
the solution obtained by using the method of characteristics with the numerical solution using
an approximation of the flow function given by Godunov method is also presented. This work
has partly been inspired by the construction of solutions of the problem with the method of
characteristics by Anestis [1] (see also [2]). Furthermore, numerical solutions under slightly
different assumptions on the flux function, are presented in [3].
Joint work with:
Raimund Bürger1 , CI2 MA and Departamento de Ingeniería Matemática, Facultad de Ciencias
Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile.
Stefan Diehl2 , Centre for Mathematical Sciences, Lund University, Lund, Sweden.
∗ Partially supported by CONICYT (Chile) through projects; BASAL project CMM, Universidad de Chile and Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción; Anillo ACT1118 (ANANUM),
e-mail: [email protected]
1 Partially supported by CONICYT (Chile) through projects Fondecyt 1130154; BASAL project CMM, Universidad de Chile and Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción;
Anillo ACT1118 (ANANUM); CRHIAM, project CONICYT/FONDAP/15130015; and Fondef ID15I10291, e-mail:
[email protected]
2 Partially supported by Lund University, e-mail: [email protected]
1
References
[1] G. Anestis, Eine eindimensionale Theorie der Sedimentation in Absetzbehältern veränderlichen Quersch-nitts und in Zentrifugen. PhD Thesis, TU Vienna, Austria, 1981.
[2] G. Anestis, W. Schneider, Application of the theory of kinematic waves to the centrifugation
of suspensions. Ing. arch., vol 53, pp. 399–407, (1983).
[3] R. Bürger, J. J. R. Damasceno, K. H. Karlsen, A mathematical model for batch and
continuous thickening of flocculated suspensions in vessels with varying cross-section. Int. J.
Miner. Process., vol. 73, pp. 183–208, (2004).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A priori and a posteriori error analyses of a
flux-based mixed-FEM for
convection-diffusion-reaction problems
Jessika Camaño∗
Departamento de Matemática y Física Aplicadas
Universidad Católica de la Santísima Concepción
Concepción, Chile, and
2
CI MA, Universidad de Concepción,
Concepción, Chile.
Abstract
In this work we propose and analyze a new mixed finite element method for the diffusionconvection-reaction problem with non-homogeneous Dirichlet boundary conditions. We consider
a mixed formulation, which yields the flux as the main unknown of the system. The original
unknown u is easily recovered as a simple postprocess of its gradient. Then, we apply the
Generalized Lax-Milgram Lemma to derive sufficient conditions for the unique solvability of the
resulting continuous and discrete formulations. In particular, a feasible choice of subspaces is
given by Raviart-Thomas of order k ≥ 0 for the gradient of u. Next, we derive a reliable and
efficient residual-based a posteriori error estimator for the problem. The proof of reliability
makes use of the global inf-sup condition, Helmholtz decomposition, and local approximation
properties of the Clément interpolant and Raviart-Thomas operator. On the other hand, inverse
inequalities, the localization technique based on element-bubble and edge-bubble functions, and
known results from previous works, are the main tools for proving the efficiency of the estimator.
Finally, some numerical results illustrating the good performance of the method and the capability of the corresponding adaptive algorithm to localize the singularities of the solution, and
confirming the theoretical rate of convergence and the theoretical properties of the estimator,
are reported.
Joint work with:
Luis F. Gatica1 , Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile, and CI2 MA, Universidad
de Concepción, Casilla 160-C, Concepción, Chile.
Ricardo Oyarzúa2 , GIMNAP-Departamento de Matemática, Universidad del Bio-Bío, Casilla 5-C,
Concepción-Chile, and CI2 MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
∗ Partially supported by CONICYT-Chile through project Inserción de Capital Humano Avanzado en la Academia
79130048 and project Fondecyt 11140691, e-mail: [email protected]
1 Partially supported by Dirección de Investigación, Universidad Católica de la Santísima Concepción through the
project DIN 14/2016, e-mail: [email protected]
2 Partially supported by CONICYT-Chile through project Anillo ACT1118 (ANANUM), project Fondecyt 1161325,
and DIUBB project 120808 GI/EF., e-mail: [email protected]
1
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A computational approach to a spatio-temporal
and gender-structured model for hantavirus
infection in rodents
Elvis Gavilán
CI2 MA and Departamento de Ingeniería Matemática
Universidad de Concepción
Concepción, Chile
Abstract
Hantavirus represents a health problem in Chile. In particular the transmission dynamics of
this virus among rodents, has not been studied sufficiently. In this presentation, we will present
a preliminary model of the spatio temporal transmission in a gender structured rodent population. The purpose of this work is to take a deterministic model and apply some ideas of a
predator prey model [3] and we utilize a spatio-temporal version of the gender-structured model
for hantavirus infection of [2]. The non-linear system consists of a non-local conservation law for
male-gender coupled with a parabolic equation for female-gender. The non-local conservation
law describes the movement of the males that can be directed toward region with high female
density, and in the direction opposite to region with high male density.
Joint work with:
R. Bürger1 , Universidad de Concepción, Concepción, Chile.
G. Chowell, Georgia State University, Atlanta, Georgia, USA.
Pep Mulet, Universitat de València, València, Spain.
Luis-Miguel Villada, Universidad del Bío-Bío, Concepción, Chile.
References
[1] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics.
Proc. Roy. Soc. A, vol. 115, pp. 700–721, (1927).
[2] L.J.S. Allen, R.K. McCormack, C.B. Jonsson, Mathematical models for hantavirus infection in rodents. Bull. Math. Biol., vol. 68, pp. 511–524, (2006).
[3] R.M. Colombo, E. Rossi, Hyperbolic predators versus parabolic preys. Commun. Math. Sci.,
vol. 13, pp. 369–400, (2015).
1 This work was funded by CONICYT (Chile) through projects Fondecyt 1130154; BASAL project CMM, Universidad de Chile and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; Anillo
ACT1118 (ANANUM); CRHIAM, project CONI- CYT/FONDAP/15130015; and Fondef ID15I10291 (to R.B.) and
CONICYT scholarship (to C.M.)., e-mail: [email protected]
1
[4] E. Rossi, V. Schleper, Convergence of a numerical scheme for a mixed hyperbolic-parabolic
system in two space dimensions. ESAIM Math. Modelling Numer. Anal., vol. 50, pp. 475–497.
[5] S. Boscarino, R. Bürger, P. Mulet, G. Russo, L. M. Villada, Linearly implicit IMEX
Runge-Kutta methods for a class of degenerate convection-diffusion problems. SIAM J. Sci.
Comput., vol. 37(2), pp. B305–B331, (2015).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A posteriori error analysis of a fully-mixed
formulation for the Navier–Stokes/Darcy
coupled problem with nonlinear viscosity
Gabriel N. Gatica∗
CI MA and Departamento de Ingeniería Matemática
Universidad de Concepción, Casilla 160-C
Concepción, Chile
2
Abstract
In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier–Stokes equations (with nonlinear viscosity) and
the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite element subspaces employed
are piecewise constants, Raviart–Thomas elements of lowest order, continuous piecewise linear
elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the
fluid, respectively, whereas Raviart–Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are
considered in the porous medium. The proof of reliability of the estimator relies on a global
inf-sup condition, suitable Helmholtz decompositions in the fluid and the porous medium, the
local approximation properties of the Clément and Raviart–Thomas operators, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based
on bubble functions, and known results from previous works, are the main tools yielding the
efficiency estimate. Finally, several numerical results confirming the properties of the estimator
and illustrating the performance of the associated adaptive algorithm are reported.
Joint work with:
Sergio Caucao1 , CI2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Ricardo Oyarzúa2 , GIMNAP-Departamento de Matemática, Universidad del Bio-Bío, Casilla 5-C,
Concepción, Chile, and CI2 MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
∗ Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and project Anillo
ACT1118 (ANANUM); and by Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción, e-mail: [email protected].
1 Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and the BecasChile Programme for Chilean students, e-mail: [email protected]
2 Partially supported by CONICYT-Chile through projects Fondecyt 1161325, project Anillo ACT1118
(ANANUM) and by Dirección de Investigación Universidad del Bío-Bío, through project 120808 GI/EF, e-mail:
[email protected]
1
References
[1] J. Camaño, G.N. Gatica, R. Oyarzúa, G. Tierra, An augmented mixed finite element
method for the Navier-Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54 (2016),
no. 2, 1069–1092.
[2] S. Caucao, G.N. Gatica, R. Oyarzúa, and I. Šebestová, A fully-mixed finite element
method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity. J. of Num. Math.,
to appear.
[3] P. Clément, Approximation by finite element functions using local regularisation. RAIRO
Modélisation Mathématique et Analyse Numérique 9 (1975), 77–84.
[4] A.I. Garralda-Guillém, G.N. Gatica, A. Márquez, and M. Ruiz-Galán, A posteriori
error analysis of twofold saddle point variational formulations for nonlinear boundary value
problems. IMA J. Numer. Anal. 34 (2014), no. 1, 326–361.
[5] G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method: Theory and Applications. SpringerBriefs in Mathematics. Springer, Cham, 2014.
[6] G.N. Gatica, A note on stable Helmholtz decompositions in 3D. Preprint 2016–03, Centro de
Investigación de Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile, (2016).
[7] G.N. Gatica, A. Márquez, R. Oyarzúa, and R. Rebolledo, Analysis of an augmented
fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media. Comput.
Methods Appl. Mech. Engrg. 270 (2014), no. 1, 76–112.
[8] G.N. Gatica, G. Tierra, and R. Ruiz-Baier, A posteriori error analysis of an augmented
mixed method for the Navier-Stokes equations with nonlinear viscosity. Preprint 2016–11, Centro
de Investigación de Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile, (2015).
[9] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, 5. Springer–Verlag, Berlin,
1986. x+374 pp.
[10] R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2013. xx+393
pp.
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
An augmented stress-based mixed finite element
method for the Navier-Stokes equations
with variable viscosity
Ricardo Oyarzúa∗
GIMNAP-Departamento de Matemática,
Universidad del Bío-Bío,
Concepción, Chile, and
2
CI MA, Universidad de Concepción,
Concepción, Chile
Abstract
A new stress-based mixed variational formulation for the 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 the present
case, and besides remarking that the strain-dependence for the viscosity yields a physically
more meaningful model, we notice that in order 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 since 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 and 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 employed 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 behaviour of the adaptive
algorithm, are reported.
∗ Partially supported by CONICYT-Chile through projects Fondecyt 1161325, project Anillo ACT1118
(ANANUM) and by Dirección de Investigación Universidad del Bío-Bío, through project 120808 GI/EF, e-mail:
[email protected]
1
Joint work with:
Jessika Camaño1 , Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile, and CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Gabriel N. Gatica2 , CI2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Ricardo Ruiz-Baier3 , Mathematical Institute, Oxford University, Andrew Wiles Building, Woodstock Road, OX2 6GG Oxford, UK.
References
[1] J. Camaño, G.N. Gatica, R. Oyarzúa, G. Tierra, An augmented mixed finite element
method for the Navier-Stokes equations with variable viscosity. SIAM J. Numer. Anal. 54 (2016),
no. 2, 1069–1092.
[2] J. Camaño, R. Oyarzúa, G. Tierra, Analysis of an augmented mixed-FEM for the NavierStokes problem. Math. Comp., DOI: http://dx.doi.org/10.1090/mcom/3124.
[3] G.N. Gatica, A note on stable Helmholtz decompositions in 3D Preprint 2016-03, Centro de
Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile, (2016).
available at http://www.ci2ma.udec.cl/publicaciones/prepublicaciones.
[4] G.N. Gatica, R. Ruiz–Baier, and G. Tierra., A posteriori error analysis of an augmented
mixed method for the Navier–Stokes equations with nonlinear viscosity. Preprint 2016-11, Centro
de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile, (2016).
available at http://www.ci2ma.udec.cl/publicaciones/prepublicaciones.
1 Partially supported by CONICYT-Chile through project Inserción de Capital Humano Avanzado en la Academia
79130048 and project Fondecyt 11140691, e-mail: [email protected]
2 Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, project Anillo
ACT1118 (ANANUM), e-mail: [email protected]
3 e-mail: [email protected].
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
Finite element analysis of a pressure-stress
formulation for the time-domain
fluid-structure interaction
Carlos García∗
CI MA and Departamento de Ingeniería Matemática
Universidad de Concepción,
Concepción, Chile
2
Abstract
We present a convergence analysis for the space discretization of a time-dependent system
of partial differential equations modeling an elasto-acoustic interaction problem. We use the
Arnold-Falk-Winther mixed finite element method with weak symmetry in the solid and the
usual Lagrange finite element method in the acoustic medium. The error analysis of the resulting
global semi-discrete scheme relies essentially on the mapping properties of an adequate projector.
We show that the method is stable uniformly with respect to the space discretization parameter
and the Poisson modulus and we prove asymptotic error estimates.
Joint work with:
Gabriel N. Gatica1 , Centro de Investigación en Ingeniería Matemática CI2 MA and Departamento
de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile.
Salim Meddahi2 , Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo,
Oviedo, España.
References
[1] C. García, G. N. Gatica, S. Meddahi, A new mixed finite element analysis of the elastodynamic equations. Applied Mathematics Letters, vol. 59, pp. 48–55,(2016).
[2] G. N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham, (2014).
∗ Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile; by project Anillo
ACT1118 (ANANUM), e-mail: [email protected]
1 Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile; by project Anillo
ACT1118 (ANANUM), e-mail: [email protected]
2 Partially supported by Ministery of Education of Spain through the project MTM2013-43671-P, e-mail:
[email protected]
1
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
Analysis of a HDG method applied to n-dimensional
linear Brinkman models
Luis F. Gatica∗
Departamento de Matemática y Física Aplicadas
Universidad Católica de la Santísima Concepción
and
Centro de Investigación en Ingeniería Matemática (CI2 MA)
Universidad de Concepción,
Concepción, Chile
Abstract
In this talk we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method
for the linear Brinkman model of porous media flow in two and three dimensions, with nonhomogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the
main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas
the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based
error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable
and efficient residual-based a posteriori error estimator, and propose the associated adaptive
algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the
expected behaviour of the adaptive refinements, are presented.
Joint work with:
Filánder A. Sequeira1 , Escuela de Matemática, Universidad Nacional de Costa Rica, Heredia,
Costa Rica.
References
[1] B. Cockburn, J. Gopalakrishnan, N. C. Nguyen, J. Peraire and F. J. Sayas, Analysis
of HDG methods for Stokes flow. Math. Comp., 80 (2011) pp. 723-760.
[2] B. Cockburn, J. Gopalakrishnan and F. J. Sayas, A projection-based error analysis of
HDG methods. Math. Comp., 79 (2010) pp. 1351-1367.
∗ Partially supported by Dirección de Investigación of the Universidad Católica de la Santísima Concepción, through
the project DIN 14/2016, e-mail: [email protected]
1 e-mail: [email protected]
1
[3] G.N. Gatica, A note on stable Helmholtz decompositions in 3D. Preprint 2016-03, Centro de
Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile.
[4] G. N. Gatica and F. A. Sequeira, Analysis of an augmented HDG method for a class of
quasi-Newtonian Stokes flows. J. Sci. Comput., 65 (2015), pp. 1270-1308.
[5] G. N. Gatica and F. A. Sequeira, Analysis of the HDG method for the Stokes-Darcy
coupling. Preprint 2015-23, Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de Concepción, Chile.
[6] G. N. Gatica and F. A. Sequeira, A priori and a posteriori error analyses of an augmented
HDG method for a class of quasi-Newtonian Stokes flow. J. Sci. Comput., to appear.
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A posteriori error analysis of an augmented
fully–mixed FEM for the Boussinesq problem
Eligio Colmenares∗
Centro de Investigación en Ingeniería Matemática (CI2 MA) and
Departamento de Ingeniería Matemática
Universidad de Concepción,
Concepción, Chile
Abstract
In this talk we present an a posteriori error analysis for a high–order quasi–optimally convergent
augmented fully–mixed finite element method introduced and analyzed in an earlier work of us to
numerically simulate heat driven flows in the Boussinesq approximation setting. Our approach
there incorporates as additional unknowns a modified pseudostress tensor and an auxiliary vector in the governing fluid and heat equations, respectively, the pressure is then eliminated by its
own definition, and redundant Galerkin terms are included to the resulting weak formulation.
The corresponding solvability analysis, its discretization, and the convergence of the latter were
stated, and for any conforming family of finite element subspaces. Optimal order a priori error
estimates were particularly proven by using Raviart–Thomas elements for the aforementioned
auxiliary unknowns, and Lagrange elements for the velocity and the temperature. Here we
propose a reliable and efficient, fully local and computable, residual–based a posteriori error
estimator in two and three dimensions. Standard arguments based on duality techniques, stable
Helmholtz decompositions, and well–known results from previous a posteriori error analyses of
related mixed schemes are the main underlying tools used in our methodology. Numerical experiments validate the expected behavior of the associated adaptive algorithm and illustrate the
accuracy improvement of the technique for approximating not only the principal unknowns but
also several other physically relevant post-processed variables, such as the pressure, the vorticity
fluid, the shear–stress tensor, and the velocity and the temperature gradient.
Joint work with:
Gabriel N. Gatica1 , CI2 MA y Departmento de Ingeniería Matemática, Universidad de Concepción, Concepción, Chile.
Ricardo Oyarzúa2 , GIMNAP-Departamento de Matemática, Universidad del Bío-Bío y CI2 MA –
Universidad de Concepción, Concepción, Chile.
∗ Partially supported by the Becas–Chile programme for foreign students and Centro de Investigación en Ingeniería
Matemática (CI2 MA), Universidad de Concepción, e-mail: [email protected]
1 Partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, project Anillo
ACT1118 (ANANUM), and project Fondecyt 11121347, e-mail: [email protected]
2 Partially supported by Universidad del Bío-Bío through DIUBB project 120808 GI/EF, e-mail:
[email protected]
1
References
[1] M. Álvarez, G. N. Gatica, R. Ruíz–Baier, A posteriori error analysis for a viscous
flow–transport problem. ESSAIM: Mathematical Modelling and Numerical Analysis, DOI:
http://dx.doi.org/10.1051/m2an/2016007.
[2] E. Colmenares, G. N. Gatica, R. Oyarzúa, An augmented fully-mixed finite element
method for the stationary Boussinesq problem. Calcolo, to appear. DOI: 10.1007/s10092-0160182-3.
[3] E. Colmenares, G. N. Gatica, R. Oyarzúa, Analysis of an augmented mixed–primal formulation for the stationary Boussinesq Problem. Numerical Methods for Partial Differential
Equations, vol. 32, 2, pp. 445-478, (2016).
[4] E. Colmenares, G. N. Gatica, R. Oyarzúa, Fixed point strategies for mixed variational
formulations of the stationary Boussinesq problem. Comptes Rendus - Mathematique, vol. 354,
1, pp. 57-62, (2016).
[5] E. Colmenares, M. Neilan, Dual–mixed formulations for the stationary Boussinesq problem.
Preprint 2016–07, Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de
Concepción, Chile, (2016).
[6] C. Dominguez, G. N. Gatica, S. Meddahi, A posteriori error analysis of a fully-mixed finite
element method for a two-dimensional fluid-solid interaction problem. Journal of Computational
Mathematics, vol. 33, 6, pp. 606-641, (2015).
[7] G. N. Gatica, G. Hsiao, S. Meddahi, A residual–based a posteriori error estimator for
a two-dimensional fluid-solid interaction problem. Numerische Mathematik, vol. 114, 1, pp.
63-106, (2009).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A Schwarz algorithm in H(curl) for irregular
subdomains in 3D
Juan Gabriel Calvo ∗
Centro de Investigación en Matemática Pura y Aplicada
Universidad de Costa Rica
San José, Costa Rica
Abstract
A new coarse space for a two-level overlapping Schwarz algorithm is presented for 3D problems
posed in H(curl). Previous studies [1, 2] for these methods are very restrictive about the geometry of the subdomains, and this new space is valid 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. In most of the existing
domain decomposition literature, the subdomains are assumed to be tetrahedra or cubes, or the
union of a few such objects or to be at least convex. The coarse space is often assumed to be
the Nédélec space on such a special coarse triangulation. Subdomains can be quite irregular for
example if they are obtained from a mesh partitioner, and there is no straightforward approach
to define coarse functions for such subdomains. In practice it is also normal to have discontinuities in the material properties, so it is very restrictive to assume that the coefficients are
constant. The goal is to develop an algorithm that can be defined for any subdomain geometry
and that works for highly discontinuous coefficient distributions. Numerical experiments with
irregular subdomains and different coefficient distributions are presented. These results are
very promising, even for random and discontinuous values of the coefficients. The algorithm is
scalable, independent of the number of degrees of freedom on each subdomain and independent
of discontinuities in the coefficients. Numerically, the condition number of the preconditioned
system grows quadratically as a function of H/δ, similar to the bound obtained in [1, 2] for the
particular case of regular subdomains and constant coefficients.
References
[1] A. Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions, Numer.
Math. vol. 86, pp. 733–752, (2000).
[2] R. Hiptmair and A. Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of Partial Differential Equations
(P. Bjørstad and M. Luskin, eds.),of IMA Vol. Math. Appl., Springer, vol. 120, pp. 181–208,
(2000).
∗ e-mail:
[email protected]
1
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
High order numerical schemes for one-dimension
non-local conservation laws
Luis-Miguel Villada Osorio∗
GIMNAP-Departamento de Matemticas
Universidad del Bío-Bío
Concepción, Chile
and
2
CI MA, Universidad de Concepción,
Concepción, Chile
Abstract
This talk deals the numerical approximation of the solutions of scalar conservation laws with
non-local flux. This equations can be applied to models of traffic flow [3] in which the mean
velocity depends on a weighted mean of the downstream traffic density or sedimentation models
[1] where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood. In both models, velocity is a function of a convolution term between
the unknown and a finite supported kernel function. The solution of this equations can exhibit
oscillations that are very difficult to approximate using classical first order numerical schemes.
We propose to use Discontinuous Galerkin methods [2] and Finite Volume WENO scheme [4]
to obtain high order approximations. DG methods can be applied in a natural way, however
their CFL restriction is very strong. FV-WENO schemes present less restrictive CFL conditions
but it is necessary to use quadratic polynomials in each cell to evaluate the convolution term in
order to obtain a high order approximation. Simulations are presented for both applications..
Joint work with:
Paola Goatin1 , INRIA Sophia Antipolis - Méditerranée, France.
Christophe Chalons2 , Université Versailles Saint-Quentin-en-Yvelines, France .
References
[1] F. Betancourt, R. Bürger, K. Karlsen, E. Tory, On nonlocal conservation laws modelling sedimentation. Nonlinearity, 24, pp. 855–885, (2011).
[2] B. Cockbur, C-W. Shu, Runge-Kutta Discontinuous Galerkin methods for convectiondominate problems. J. Sci. Comput., vol. 16, pp. 173–261, (2001).
∗ Partially
supported by Fondecyt project 11140708 , e-mail: [email protected]
[email protected]
2 e-mail: [email protected]
1 e-mail:
1
[3] P. Goatin, S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic
flow model with non-local velocity. Netw. Heterog. Media, vol. 11(1), pp. 107–121, (2016).
[4] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for
hyperbolic conservation laws. Springer Berlin Heidelberg, pp. 325–432, (1998).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A priori and a posteriori error analysis of
an augmented mixed-FEM for the
Navier-Stokes/Brinkman problem
Nestor Sánchez∗
GIMNAP-Departamento de Matemática,
Universidad del Bío-Bío,
Concepción, Chile
Abstract
We introduce and analyze an augmented mixed finite element method for the Navier-StokesBrinkman problem. We employ a technique previously applied to the stationary Navier-Stokes
equation, which consists of the introduction of a modified pseudostress tensor relating the gradient and the pressure with the convective term, and propose a pseudostress-velocity formulation
for the model problem. Since the convective term forces the velocity to live in a smaller space
than usual, we augment the variational formulation with suitable Galerkin type terms. The
resulting augmented scheme is then written equivalently as a fixed point equation, so that the
well-known Banach theorem, combined with the Lax-Milgram theorem, are applied to prove the
unique solvability of the continuous and discrete systems. We point out that 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, Raviart-Thomas spaces of order k and continuous piecewise polynomials of degree
≤ k +1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. In addition, we derive a reliable and efficient residual-based
a posteriori error estimator for the augmented mixed method. The proof of reliability makes
use of the global inf-sup condition, a Helmholtz decomposition, and local approximation properties of the Clément interpolant and Raviart-Thomas operator. On the other hand, inverse
inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the L2 -orthogonal projector, and known results from previous works,
are the main tools for proving the efficiency of the estimator. Finally, several numerical results
illustrating the performance of the augmented mixed method, confirming the theoretical rate of
convergence and the theoretical properties of the estimator, and showing the behaviour of the
associated adaptive algorithms, are reported.
Joint work with:
Luis F. Gatica1 , Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile,
∗ Partially supported by CONICYT-Chile through projects Fondecyt 11121347, project Anillo ACT1118
(ANANUM) and by Dirección de Investigación Universidad del Bío-Bío, through project 120808 GI/EF, e-mail:
[email protected]
1 Partially supported by Dirección de Investigación, Universidad Católica de la Santísima Concepción through
project DIN 14/2016, e-mail: [email protected]
1
and CI2 MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Ricardo Oyarzúa2 , GIMNAP-Departamento de Matemática, Universidad del Bio-Bío, Casilla 5-C,
Concepción, Chile, and CI2 MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
References
[1] J. Camaño,
R. Oyarzúa and G. Tierra,
Analysis of an augmented
mixed-FEM for the Navier-Stokes problem. Mathematics of Computation, DOI:
http://dx.doi.org/10.1090/mcom/3124
[2] S. Caucao, D. Mora and R. Oyarzúa, A priori and a posteriori error analysis of a
pseudostress-based mixed formulation of the Stokes problem with varying density. IMA Journal of Numerical Analysis, vol. 36, 2, pp. 947-983, (2016).
[3] G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity
allowing RT0 –P1 –P0 approximations. ESAIM: Mathematical Modelling and Numerical Analysis,
vol. 40, 1, pp. 1–28, (2006).
[4] G.N. Gatica, A note on stable Helmholtz decompositions in 3D. Preprint 2016-03, Centro de
Investigación en Ingeniería Matemática (CI2MA), UDEC, (2016).
[5] G.N. Gatica, L.F. Gatica and A. Márquez, Analysis of a pseudostress-based mixed finite
element method for the Brinkman model of porous media flow. Numerische Mathematik, vol
126, 4, pp. 635-677, (2014).
2 Partially supported by CONICYT-Chile through projects Fondecyt 11121347 and 1161325, project Anillo
ACT1118 (ANANUM) and by Dirección de Investigación Universidad del Bío-Bío, through project 120808 GI/EF,
e-mail: [email protected]
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
A high order HDG method for Stokes flow
in curved domains
Felipe Vargas Martínez∗
CI2 MA and Departmento de Ingeniería Matemática,
Universidad de Concepción,
Concepción, Chile
Abstract
We propose and analyze a high order hybridizable discontinuous Galerkin (HDG) method for
the Stokes equations in a curved domain Ω. It is based on approximating Ω by a polygonal/polyhedral domain where an HDG approximation can be computed. In in order to obtain
a suitable approximation for the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the extrapolated discrete gradient. We also
propose to extrapolate the discrete pressure and impose its mean value over the computational
domain in such a way that the approximated pressure has zero mean in the entire domain Ω.
We show that, if the computational domain is defined through interpolating the boundary of
Ω by a piece-wise linear function, the method provides optimal order of convergence, i.e., order
k + 1 for the approximations of the pressure, the velocity and its gradient; and order k + 2 for
the numerical trace of the velocity and for the element-by-element post-processed velocity. We
also provide numerical experiments validating the theoretical error estimates.
Joint work with:
Manuel Solano1 , Departmento de Ingeniería Matemática & CI2 MA, Universidad de Concepción,
Concepción, Chile.
References
[1] B. Cockburn, J. Gopalakrishnan, N. C. Nguyen, J. Peraire, Analysis of HDG methods
for Stokes flow. Math. of Comp., vol. 80 (274), pp. 723–760 (2010).
[2] B. Cockburn, J. Gopalakrishnan and F.-J. Sayas, A projection-based error analysis of
HDG methods. Math. Comp., vol. 79, pp. 1351–1367, (2010).
[3] B. Cockburn, W. Qiu and M. Solano, A priori error analysis for HDG methods using
extensions from subdomains to achieve boundary-conformity. Math. of Comp. vol. 83, pp. 665–
699 (2014).
∗ Partially
supported by the Scholarship Program of CONICYT-Chile, e-mail: [email protected]
supported by CONICYT-Chile through the FONDECYT project No. 1160320, BASAL project CMM,
Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (CI2 MA), 24 Universidad de Concepción,
and by Project Anillo ACT1118 (ANANUM), e-mail: [email protected]
1 Partially
1
[4] M. Solano and F. Vargas, A high order HDG method for Stokes flow in curved domains.
Preprint 2016-12, Centro de Investigación en Ingeniería Matemática (CI2 MA), Universidad de
Concepción, 2016.
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y 5 de Agosto de 2016, Antofagasta, Chile
Métodos de alto orden para sistemas hiperbólicos
con productos no conservativos, aplicados a sistemas
shallow water multicapa con sedimentación
polidispersa
Víctor Osores∗
CI2 MA and Departamento de Ingeniería Matemática,
Universidad de Concepción,
Concepción, Chile
Abstract
En este trabajo consideramos una mezcla formada por un fluido viscoso y material particulado
disperso en ella formado principalmente por partículas sólidas pequeñas de diferentes especies
y diferentes tamaños. El modelo que se presenta nace de combinar un sistema Shallow Water
(o Saint Venant) con un sistema de sedimentación polidispersa, esto permite conocer tanto el
comportamiento vertical de la especies en la mezcla como también el movimiento horizontal
de ellas en el fluido. El modelo resultante logra ser escrito en forma condensada como un sistema hiperbólico con productos no conservativos, el cual resolvemos a través técnicas de tipo
volúmenes finitos para sistemas hiperbólicos con este tipo de productos. Se muestra como a
partir de un método numérico a orden uno, conocida la función de flujo numérico F (U, V ), es
posible desarrollar métodos de orden mayor para este tipo de ecuaciones. Aquí no consideramos el efecto de sedimento compresible, ni tampoco la variación de la batimetría producto del
sedimento depositado en el fondo.
Trabajo conjunto con:
Raimund Bürger, CI2 MA, Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas
y Matemáticas, Universidad de Concepción, Concepción, Chile.
Enrique Fernández-Nieto, Departamento de Matemática Aplicada I, E.T.S. Arquitectura, Universidad de Sevilla, Sevilla, España.
References
[1] E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation. Discrete
Contin. Dyn. Syst. Ser. B, vol. 5, pp. 189–214 (2005).
[2] S. Berres, R. Bürger, K.H. Karlsen, E.M. Tory, Strongly degenerate parabolic-hyperbolic
systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math., vol. 64,
pp. 41–80, (2003).
∗ Financiado parcialmente por CONICYT (Chile) a traves de proyecto CMM basal, Anillo ACT118 (ANANUM),
e-mail: [email protected]
1
[3] M. Castro, J. M. Gallardo, C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to
shallow-water systems. Math. Comput., vol. 1103, (1134), pp. 281–314 .
[4] E.D. Fernández-Nieto, E.H. Koné, T. Morales de Luna, R. Bürger. A multilayer
shallow water system for polydisperse sedimentation. J. Comput. Phys., vol. 238, pp. 281–314,
(2013).
2
XXV COMCA
Congreso de Matemática Capricornio
2,3,4 y de Agosto de 2016, Antofagasta, Chile
A new sequential algorithm for fluid mechanics and
heat transfer in complex conjugate problems solved
by finite volume method
Nelson Moraga ∗
Departamento de Ingeniería Mecánica
Universidad de La Serena
Benavente 980, La Serena, Chile
Abstract
A novel prediction-corrector algorithm to solve the Navier-Stokes equations coupled to the unsteady heat convection-diffusion equations for incompressible flows is presented. The objective
of this work is to analyze the effects on accuracy and numerical efficiency of using inner doublyiterative processes [1], followed by two cycles of prediction-correction for solving the continuity,
linear momentum and energy equations [2]. The performance of the new algorithm for the Finite Volume Method on solving unsteady natural heat convection problems inside square and
annular cylindrical thick walled cavities is investigated. Increments in accuracy is accounted
for by comparison with experimental available data of air and water convection for Rayleigh
numbers Ra = 104 and 105 . Enhancement on robustness is described by the influence of the
under-relaxation coefficients for the velocity components on the number of iterations in the solution of natural convective heat problems. Efficiency in the solution procedure is examined by
the computation time required with the new algorithm in relation to the classical SIMPLE [3]
and PISO [4] algorithms.
Joint work with:
Juan Jaime, Departamento de Ingeníería Mecánica, Universidad de La Serena, Benavente 980, La
Serena, Chile.
References
[1] D.I. Sun, Z.Q. Qu, Y.L. He, W.Q. Tao, An efficient segregated algorithm for incompressible fluid flow and heat transfer problems-IDEAL (inner doubly-iterative efficient algorithm
for linked-equations) part I: mathematical formulation and solution procedure. Numerical Heat
Transfer B, vol. 53, pp. 1–17, (2008).
[2] N.O.Moraga, S.C. Ramírez, M.J. Godoy, P. Ticchione, Study of convective nonNewtonian alloy solidification in molds by the Psimpler/Finite volume method. Numerical Heat
Transfer A, vol. 12, pp. 936–953, (2010).
∗ The authors acknowledge the support received from CONICYT-Chile to FONDECYT 1140074 project, e-mail:
[email protected]
1
[3] S.V. Patankar, B. Spalding, A calculation procedure for heat, mass and momentum transfer
in three-dimensional parabolic flows. International Heat Mass Transfer vol. 15, pp. 1787-1806,
(1972).
[4] R.I. Issa, Solution of implicitly discretized fluid flow equation by operator splitting. J. Computational Physics, vol. 62, pp. 40–65 (1985).
2