### Researchers

U. Concepción: Rodolfo Araya, Raimund Bürger, Rommel Bustinza, Gabriel N. Gatica, Rodolfo Rodríguez, Mauricio Sepúlveda.

UTFSM: Erwin Hernández

### Main Collaborators

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).

## About the research group

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.

### Numerical methods in transient electromagnetics

The development and analysis of a numerical framework to deal with transient electromagnetics is a need for modeling several industrial applications. Among them we mention the electromagnetic forming, which is a metallurgical working process based on the use of electromagnetic forces to deform metallic work pieces at high speeds. One of the main challenges in the analysis of the underlying mathematical models is the fact that it involves equations posed on moving (conductor) domains.

Another example of transient electromagnetics applications is the analysis and design of electrical machines. In this field, an important challenge is the accurate computation of the power losses in the ferromagnetic components of the core. In particular hysteresis losses, which are related to the intrinsic nature of magnetic materials and whose modeling involves highly non-linear equations, since the material behavior depends on its magnetization history.