Researchers DIM - U de Concepción
Rodolfo Araya, Raimund Bürger, Gabriel N. Gatica, Rodolfo Rodríguez, Mauricio Sepúlveda.

Postdoctorals
Past: Fabrice Jaillet (France), Franz Choully (France), Stefan Berres (Germany), Tatiana Voitovich (Germany), María González (Spain), Luis Hervella-Nieto (Spain).

Ph.D. Students
Current: Ramiro Acevedo, Fernando Betancourt, Emilio Cariaga, David Mora, Ricardo Oyarzua, Abner Poza, Carlos Reales, Ricardo Ruiz, Frank Sanhueza, Héctor Torres, Carlos Vega.
Past: Mauricio Barrientos, Tomás Barrios, Aníbal Coronel, Galina García, Luis Gatica, Erwin Hernández, Mario Mellado, Cristian Pérez, Edwin Behrens, Rommel Bustinza, Antonio García.

Mathematical Engineering Thesis
Current: Leonardo Figueroa, Cristian Muñoz, Manuel Solano.
Past: Daniel Arroyo, Pablo Durán, Karina Malla, Ricardo Monge, Mario Quijada, Marcela Torrejón.

Main Collaborators
Rommel Bustinza (Chile), Gabriel R. Barrenechea (United Kingdom), Alfredo Bermúdez (Spain), Bernardo Cockburn (USA), Ricardo Durán (Argentina), Hermano Frid (Brazil), Norbert Heuer (United Kingdom), Kenneth H. Karlsen (Norway), Salim Meddahi (Spain), Nil-Henrik Risebro (Norway), Pilar Salgado (Spain), Francisco-Javier Sayas (Zaragoza), Kai Schneider (France), John Towers (USA), Frédéric Valentin (Brazil).

Our group has made important advances in the mathematical modeling and numerical solution of various physical problems arising in potential theory, electromagnetism, elasticity, fluid mechanics, fluid-solid interactions, acoustics, hyperbolic conservation laws, sedimentation processes, wave propagation, oceanography and environmental flows.

Our current research interests are:

Mixed finite element methods (FEM) for nonlinear problems in mechanics
In elasticity and fluid mechanics we study a posteriori error estimates for augmented and stabilized mixed FEM applied to advection-diffusion-reaction, Stokes, Navier-Stokes and elasticity problems. For incompressible flows in porous media we study a priori and a posteriori error analysis of mixed FEMs for velocity-pressure-stress formulations of incompressible flows. Our approach differs from the standard one which, instead of the full stress, introduces the extra stress tensor as unknown.

Continuum mechanics and electromagnetism
The local discontinuous Galerkin method is a well established tool to solve diffusion dominated and purely elliptic equations, providing high order approximation, high parallelism, and flexibility for h, p, hp refinements. Our goal is to extend LDG methods to elasticity and transmission problems in continuum mechanics and electromagnetism.

Conservation laws and reaction-diffusion-convection problems
Conservation laws describe phenomena such as traffic flows, combustion, pattern formation, chemotaxis and mathematical ecology. We study the well-posedness and numerical analysis of time-dependent nonlinear PDEs close to conservation laws but with non-standard ingredients such as discontinuous flux, degenerate diffusion, non-local flux, network-type domains, and hyperbolic-elliptic degeneracy. For conservation laws and degenerate parabolic equations that develop solutions with sharp fronts, multi-resolution methods use wavelet-based representations to adaptively concentrate the computational effort on the areas of strong variation. Experience has been gained for one-dimensional problems, and our future efforts will be directed to extend this approach to several space dimensions.

Wave equations
The Perfectly Matched Layer method is a recent approach to truncate the computational domain of a scattering problem, by simulating an absorbing layer of damping material around the domain. Our aim is to provide a rigorous numerical analysis to explain the excellent numerical performance of the method.

Electromagnetic forming of sheet metals
Electromagnetic forming is a metal working process based on the use of electromagnetic forces to deform metallic work pieces at high speeds. A striking aspect is the lack of mathematical and numerical models that capture the pertinent physics and offer accurate solutions. This issue represents a major motivation for this research.

The members of this group, which is based at the Centro de Investigación en Ingeniería Matemática (CI²MA) of the  Universidad de Concepción, are the following: Rodolfo Araya, Raimund Bürger, Rommel Bustinza, Gabriel N. Gatica, Rodolfo Rodríguez and Mauricio Sepúlveda. The main research topics that have been addressed by them jointly with their collaborators and students, during the period of evaluation (2008 – 2012) are described below.

Conferences. During the period 2008 – 2011 we have been main organizers of several well-known meetings and conferences. Among them, we highlight the third version of our international series named WONAPDE: Third Chilean Workshop on Numerical Analysis of Partial Differential Equations, held at the Universidad de Concepción during January 11-15, 2010. Since the first of these workshops on January 2004, the events WONAPDE, which we run every three years, have seen a continuous increase of participants (about 150 for the version of 2010), and each one has been associated with the publication of a Special Issue of the ISI journal Applied Numerical Mathematics. Presently, we are already organizing the WONAPDE 2013, to be held in January 14 – 18, next year. On the other hand, we have also organized the Fourth, Fifth and Sixth National Meeting on Numerical Analysis of Partial Differential Equations, which were held in/at: January 2009/PUC, December 2010/PUC, and December 2011/U. de La Serena, respectively. In addition, some of our researchers co-organized several other international meetings held in Chile and abroad, such as Advances in Boundary Integral Equations and Related Topics (U. of Delaware, USA, August 2009), Current & New Trends in Scientific Computing (U. de Chile, October 2009), and the Symposium: Partial Differential Equations: Theory, Applications, Simulations (U. Stuttgart, Germany, October 2011). Furthermore, members of this group have organized several invited sessions and minisymposia in national and international conferences.

Training and placements. Since 2008, we have supervised 18 Ph.D. students, 12 of them already defended their thesis. Their current activities are as follows: 3 academic positions at foreign universities; 7 academic positions at national universities: U. Bio-Bio (2), U. Católica de Concepción (1), U. La Serena (1), U. de Concepción (1), U. Andres Bello (1), U. Católica de Temuco (1); 2 postdoctorates: U. British Columbia, Canada (1), EPFL, Switzerland (1). In addition, during the period 2008 – up to now, we have also supervised 12 undergraduate students, 6 of them already graduated or are next to do it, of which 4 are presently doing Ph.D. either with us (2) or abroad (1 in Brown University, 1 in U. de Paris 6).

Network. Our main collaborators from Chile and abroad are: K. H. Karlsen (U. of Oslo, Norway), S. Diehl (Lund U., Sweden), I. Nopens (Ghent U., Belgium), C. Rohde (U. of Stuttgart, Germany), R. Ruiz-Baier (EPFL, Lausanne, Switzerland), K. Schneider (Aix-Marseille Université, Marseille, France), P. Mulet (U. de Valencia, Spain), J.D. Towers (MiraCosta College, Cardiff-by-the-Sea, USA), Carlos D. Acosta (U. Nacional de Colombia, Sede Manizales), A. Pazoto (U. of Rio de Janeiro, Brasil),  M. Bendahmane (U. Bordeaux, France), V. Anaya (U. Bio-Bio, Chile), J.Muñoz-Rivera (LNCC, Brasil), S. Pop (U. of Eindhoven, The Netherlands), E. Cariaga (U. Catolica, Temuco, Chile), N. Heuer (PUC, Chile), G.C. Hsiao (U. of Delaware, USA), L. Gatica (UCSC, Chile), M. Maischak (U. Brunel, UK), A. Marquez (U. de Oviedo, Spain), S. Meddahi (U. Oviedo, Spain), R. Oyarzua (UBB, Chile), M. Sanchez (UdeC, Chile), F.J. Sayas (U. of Delaware, USA), E.P. Stephan (U. Hannover, Germany), F. Valentin (LNCC, Brasil), P. Frey (U. Paris 6, France), F. Jaillet (U. Lyon, France), G. Barrenechea (U. Strathclyde, Scotland), T. Barrios (UCSC, Chile), V. Dominguez (U. Navarra, Spain), E. Hernandez (UFSM, Chile), A. Bermúdez and P. Salgado (both, U. Santiago de Compostela, Spain), C. Lovadina and L. Beirao da Veiga (both, U. Pavia, Italy), D. Mora (U. Bio Bio, Chile), R. Durán (U. Buenos Aires, Argentina) and C. Padra (Centro Atómico Bariloche, Argentina).

Some of our main scientific achievements during the period 2008 – up to now, which are supported by approximately 94 papers in ISI journals, are described in what follows.

Multiresolution methods for the efficient solution of reaction-diffusion systems in electrocardiology and related applications. Multiresolution methods were first analyzed for one-dimensional strongly degenerate parabolic equations (M2AN 2008, J. Eng. Math. 2008), and then for spatially two-dimensional reaction-diffusion systems (Appl. Numer. Math. 2009) Including adaptive time stepping. They provide an efficient numerical tool for the simulation of problems where solutions strongly vary over small length scales. Our efforts concentrated, in particular, on models of electrocardiology (Numer. Meth. PDE 2010, J. Sci. Comput. 2010), and then moved to coupled sedimentation-flow systems (Int. J. Numer. Anal. Model. 2012). The study of reaction-diffusion systems, more precisely of a particular variant of the Keller-Segel model of chemotaxis, also motivated the well-posedness analysis in Math. Meth. Appl. Sci. (2012), while the application to electrocardiology gave rise to the convergence analysis of a (non-adaptive) finite volume method (Math. Comput. Simulation 2010). 

Well-posedness and numerical analysis for conservation laws and related equations with discontinuous flux. Conservation laws with discontinuous flux appear in a variety of application including clarifier-thickener units and hydraulic classifiers in mineral processing, traffic flow on highways with changing road surface conditions and variable numbers of lanes, and flow in heterogeneous porous media. Fairly simple numerical methods to handle problems of these kinds were proposed, tested, and in part analyzed in (J. Eng. Math. 2008, Netw. Heterog. Media 2008). The convergence of an Engquist-Osher-type numerical scheme to the properly defined entropy solution of a conservation law with discontinous flux problem was proved in (SIAM J. Numer. Anal. 2009). For a system of conservation laws, the issue of convergence to an entropy solution is studied in (Netw. Heterog. Media 2010), and a second-order scheme for a clarifier-thickener model is advanced in (Numer. Math. 2010). 

Analytical and numerical methods for multi-species kinematic flow models. Models of this kind are given by systems of nonlinear conservation laws modeling the settling of polydisperse suspensions (with particles belonging to different size classes) or multi-class traffic flow. Exact solution constructions are advanced in (Acta Mech. 2008, Appl. Math. Modelling 2009). Numerical methods approximating a continuous particle size distribution are introduced in (Math. Models Methods Appl. Sci., 2008). In (Adv. Appl. Math. Mech. 2009) the oscillatory solution behaviour in the case of loss of hyperbolicity is studied.  The parameter regions of hyperbolicity for a choice of common polydisperse sedimentation models are estimated in (SIAM J. Appl. Math. 2010, Math. Meth. Appl. Sci. 2012). This approach also gives access to the characteristic decomposition of the flux Jacobian for these models, which is the basis of the high-resolution method introduced in (J. Comput. Phys. 2011). 

Well-posedness and numerical analysis for convection-diffusion PDEs involving discrete or continuous nonlocal operators. Existing explicit numerical schemes for degenerate parabolic convection-diffusion PDEs usually suffer from severe restrictions of the time step for a given spatial discretizations. They can be made more efficient if the computational stencil is widened by the method of discrete mollification. Convergence analysis for the resulting schemes are presented in (Numer. Meth. PDE 2012) and (IMA J. Numer. Anal. to appear) for one and two space dimensions, respectively.  On the other hand, a spatially non-local degenerate parabolic PDE modeling aggregation was studied in (Commun. Math. Sci. 2011), while a conservation law with discontinuous flux modeling sedimentation is analyzed in (Nonlinearity 2011). We estimate that the analysis of methods of discrete mollification for local PDEs leads the way to establish convergence for methods for PDEs that are posed non-locally. 

Stabilized methods and applications. We proposed a new stabilized finite volume element method to solve a coupled sedimentation flow problem for a clarifier-thickener model in two space dimensions (SIAM J. Sci. Comput., in press). In addition, we unified the theory for a family of stabilized finite element methods in fluid mechanics (Appl. Numer. Math. 2009), and derived several a posteriori error estimators for the adaptive computation of the corresponding solutions (JCAM 2008). We also developed new stabilized finite element methods for the Navier-Stokes equation (SIAM J. Numer. Anal. 2012), and analyzed and implemented new a posteriori error estimators for this equation, which are used to improve the quality of the computed numerical solutions (Numer. Meth. PDE 2012). 

Applications in engineering and environmental sciences. Some of our findings have given rise to contributions to engineering and related journals, with a strong impact for applications. These include a new model and simulation method for the classification of polydisperse suspensions (Comput. Chem. Eng. 2008), the simulation of the particle size distribution in a centrifuge (Drying Technol. 2008), and a consistent modeling methodology for secondary settling tanks in wastewater treatment, which is based on the well-posedness and numerical analysis of conservation laws with discontinuous flux (Water Research 2011, Comput. Chem. Eng. 2012, Comm. Appl. Biol. Sci. 2012). Multiresolution Methods for Strongly Degenerate Parabolic PDE. We analyzed adaptive multiresolution schemes for strongly degenerate quasilinear parabolic problems in one dimension of space, with continuous flow and discontinuous. The numerical method is based on a finite volume discretization by using the Engquist-Osher scheme for the flux and explicit time steps. It then uses an adaptive multiresolution scheme based on averages in cells, to speed up CPU time and memory requirements save the finite volume scheme, whose version of the first order, as is known in advance, converges to entropy solution of the problem. In a first job (M2AN, 2008), we study thoroughly the case of strongly degenerate parabolic problems with continuous flow and boundary conditions of zero flow rate, and periodicals. Additionally, a comparative analysis of the efficiency with the classic method of finite volumes with and without multiresolution was developed. In a second work (J. Eng. Math. 2008), we considered discontinuous flows, which occurs, for instance in models of sedimentation and vehicular traffic. 

Dispersive problems (KdV, Schrödinger). We developed new numerical methods for problems of evolution with dispersive terms as equations or systems of KdV or Schrödinger type. In particular, we study a numerical method for a coupled system of KdV with moving boundary (JCAM 2008). A numerical strategy for addressing both a viewpoint of the stability analysis numerical scheme, and also from a viewpoint of implementation, consisted decouple variable by changing the terms of the third order. Second, we studied the stability of numerical schemes for the critical case of the equation of generalized Korteweg-de Vries, adding a damping term (Numer. Math. 2010). Third, we studied some numerical methods for a coupled system of Schrödinger type (Appl. Numer. Math. 2009). The strategy was to use a Crack-Nicolson type scheme, which preserves numerically the L2 norm of the solution (or energy system). This system of Schrödinger type has a second conservation law, and we found that this second law is also preserved, at least numerically. 

Population Dynamics. We studied new numerical methods for systems of reaction-diffusion with a nonlocal diffusion. We proved the well-posedness (Acta Appl. Math. 2008) and the existence of patterns formation for a model of epidemic spread of a type susceptible-infected-recovered, SIR (Disc. Cont. Dyn. Sist. B). We also showed the well-posedness, convergence of the FVM scheme, and the existence of patterns formation of systems of diffusion-reaction modeling the early propagation of a lung tumor (M3AS, 2010), and the predator-prey system in a polluted environment (Net. Het. Media 2010).

Porous media problems and applications. We proved an a priori error estimate and the corresponding rate of convergence of the finite volume method as applied to a weakly degenerate parabolic equation modeling porous media. This result was applied to diverse models on heap leaching (JCAM 2010), and, in particular, we proved the convergence of a centered vertex finite volume method for a model of copper heap leaching (M2AS 2010). On the other hand, we introduced and analyzed new finite element methods for the Stokes-Darcy problem, which models the coupling of fluid flow with porous media flow. Our results consider primal-mixed and fully-mixed approaches, including linear and nonlinear behaviors either at the fluid or the porous medium, and they refer to well-posedness of the continuous and discrete formulations, a priori error bounds, and a posteriori error estimates (to appear in IMA J. Numer. Anal., Math. Comp. 2011, Num. Meth. PDE. 2011, CMAME 2011, SIAM J. Numer. Anal. 2010, IMA J. Numer. Anal. 2009).

Finite element methods in continuum mechanics and fluid-solid interactions. We developed new pseudostress-based mixed finite element methods and related augmented schemes, including the solution of associated exact control problems, for diverse linear and nonlinear problems in continuum mechanics, particularly for Lame, Stokes, generalized Stokes, and Brinkman equations in 2D and 3D (CMAME 2012 to appear, Comm. Comp. Phys. 2012, Inter. J. Numer. Meth. Fluids 2011, CMAME 2011, 2010, JCAM 2009, CMAME 2008, SIAM J. Sci. Comput. 2008), the main advantage of them being the fact that they become stable for larger families of finite element subspaces. On the other hand, we incorporated the utilization of the recently derived Arnold-Falk-Winther elements, and introduced and analyzed new finite element schemes for a class of fluid-solid interaction problems in 3D. Our results, which refer to primal, mixed, and augmented schemes, and also to the coupling with boundary integral equation methods, include the well posedness of the resulting continuous and discrete formulations, and the derivation of a priori and a posteriori error estimates (SIAM J. Numer. Anal. to appear, IMA J. Numer. Anal. 2011, Appl. Numer. Math. 2009, Numer. Math. 2009, Adv. Comput. Math. 2009).

Finite element methods in electromagnetism.  We introduced and analyzed a new mixed finite element method for the coupled problem arising from the interaction of a time harmonic electromagnetic field with a three-dimensional elastic body (SIAM J. Numer. Anal. 2010). We applied the abstract framework developed in a recent work by A. Buffa (SIAM J. Numer. Anal. 2005), and proved that the resulting coupled scheme is uniquely solvable and convergent. In addition, we derived a new mixed finite element method for a time harmonic Maxwell problem with an impedance boundary condition (IMA J. Numer. Anal. 2012). Our main contribution here refers to the fact that we do no need to consider a vanishing tangential trace as a boundary condition but just an impedance boundary condition in a connected component of the domain. Our analysis is free from special requirements on the finite element triangulations and also any regularity result and special assumption on the geometry of the domain. 

Discontinuous Galerkin methods in continuum mechanics. We extended the applicability of a certain class of stabilized discontinuous Galerkin method to several problems in continuum mechanics, including Helmholtz, Poisson, Darcy, and Stokes equations (Numer. Math. 2012, IMA J. Numer. Anal. 2010, Preprint CI²MA 2010-25). Our results refer to both the a priori and a posteriori error analyses, and show, in particular, that the discrete inf-sup condition can be circumvented, allowing us then to consider any pair of discrete approximation spaces. In addition, we completed the analysis of the Local Discontinuous Galerkin (LDG) method to solve a Signorini type problem (J. Sci. Comput. to appear). On the other hand, we also developed a new approach for the coupling of LDG and boundary element methods (BEM) (Math. Comp. 2010), which instead of a mortar variable as in our previous work (Math. Comp. 2006), makes use of a finite element subspace whose functions are required to be continuous only on the coupling boundary.

Numerical methods in transient electromagnetics. We developed and analyzed a numerical framework to deal with transient electromagnetics in domains involving moving conductors. This research was motivated by the need of modeling electromagnetic forming, which is a metallurgical process for metal forming. The obtained results give place, among others, to the following publications: Math. Comp. 2009; IMA J Numer. Anal. 2010; IMA J Numer. Anal. (online).

Finite element methods for slender elastic structures. We proposed and analyzed finite element methods to deal with elasticity problems of slender structures that had not been treated rigurously yet (reinforced plates, buckling of plates and beams, among others). In particular we succeeded in proving optimal order error estimates for the finite element approximation of plate buckling problems, which to the best of our knowledge, was an open problem. Among the published results on this subject, we highlight the following: IMA J. Numer. Anal. 2009; Math. Comp. 2009; SIAM J. Numer. Anal. 2010; Math. Comp. 2011.

Editorial work. On the other hand, our main editorial duties during the period have been the following: Raimund Bürger: Guest Editor of a special issue (to appear) of Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM); Raimund Bürger and Rodolfo Rodriguez: Guest Editors of vol. 59 (9) Applied Numerical Mathematics (2009). Special Issue for the Second Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2007); Raimund Bürger, Gabriel N. Gatica, and Rodolfo Rodriguez: Guest Editors of vol. 62 (4) Applied Numerical Mathematics (2012), special issue for the Third Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2010); Gabriel N. Gatica: Member of the Editorial Board of Numerical Functional Analysis and Optimization; Gabriel N. Gatica: Managing Guest Editor of vol. 62 (6) Applied Numerical Mathematics (2012). Special issue for the conference Advances in Boundary Integral Equations and Related Topics; Rodolfo Rodriguez: Member of the Editorial Board of Computer Modeling in Engineering and Sciences (CMES).