Numerical Analysis of PDEs


Rodolfo Araya, Fernando BetancourtRaimund Bürger, Jessika CamañoGabriel N. Gatica, Luis Gatica, David Mora, Ricardo OyarzúaRodolfo Rodríguez, Mauricio Sepúlveda, Manuel Solano, Luis M. Villada.

Coordinators: Raimund Bürger and Jessika Camaño

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 and related methods in continuum mechanics

This group develop new mixed and primal finite element methods (FEM) for linear and nonlinear initial or boundary value problems arising from engineering sciences. In particular this group is interested in new mixed formulations in solid and fluid mechanics, as well as in coupled versions of such problems. Other methods to be studied are discontinuous Galerkin methods (DGM), hybridizable discontinuous Galerkin methods (HDG), virtual element methods (VEM) and discontinuous Petrov-Galerkin methods (DPG). Several specific applications include hot pressing of wood fiber boards and registration of deformable images in biomedicine.

Numerical methods for conservation laws and related problems

This line of research is focused on the development and mathematical analysis of numerical methods such as finite volume methods (FVM), front tracking, DGM and finite difference methods for problems defined by systems of conservation laws, whole main distinct property is the formation of discontinuous solutions (that include shocks) even from smooth initial data. A related research topic is the development of efficient implicit-explicit solvers for convection-diffusion- reaction problems with discontinuous coefficients and strong degeneracy. Furthermore, multilayer shallow water models for the flow of solid-fluid mixtures in natural aquifers and in tubes are also studied. Specific applications include simulators for mineral processing, wastewater treatment plants and multiphase flow in porous media and for the spatio-temporal spread of infectious diseases. In addition, transport PDEs that model the interaction between polymers and metal ions with applications in copper mining and chemical engineering are considered as well.

Spectral problems in computational electromagnetism

This group focuses on the development and mathematical analysis of new FEM methods for eigenvalue problems such as the computation of Beltrami fields or transmission eigenvalues. The first of these problems can be posed as an eigenvalue problem for the curl operator with suitable boundary conditions and appropriate topological restrictions. Applications that motivate interest in this topic include solar and plasma physics. Furthermore, along the same area of problems of eigenvalues for the curl operator, this group is interested in applying FEM methods to approximate the nonlinear free force fields in which the role of the eigenvector is taken over by a scalar function that depends on position. In the case of transmission eigenvalues it is possible to quantify the presence of anomalies within a homogeneous medium and to use this information to prove the physical integrity of materials. In both cases the possibilities to employ alternative techniques such as DGM and virtual element methods (VEM) are evaluated. Furthermore, reliable and efficient a posteriori error estimators for these problems are developed. This group also works on modelling, via FEM, of photovoltaic devices such as solar cells.