ROM for the solution of parabolic problems with highly concentrated moving sources

SERGIO IDELSOHN; ALEJANDRO COSIMO; ALBERTO CARDONA

Workshop; III workshop CSMA-SEMNI on Numerical techniques for nowadays highly computationally demanding challenges: meshless, MOR and beyond; 2017

The aim is to solve parabolic problems with highly concentrated moving gradients. A naive approach to tackle this kind of problems is to refine the mesh as much as needed in all the regions of the domain that are in the vicinity of the path of the moving gradient. One alternative is to adaptively refine the mesh following the moving gradient. Despite the fact that both previous options will work, both are very expensive, mainly for the fact that to use a reduced order model (ROM) does not reduce the computer time needed to obtain a solution due to the coupling of the space and time dimensions in this type of problem. The alternative proposed in this presentation was to adopt a global/local scheme, in which a moving local fine mesh describes the neighbourhood of the moving gradient and a coarse global mesh describes the analysis domain. In order to glue the two meshes a number of Lagrange multipliers are added to the boundary of the local domain. It should be noted that the boundary of the local domain does not need to conform to the elements´ boundary of the coarse global domain. The advantage of the proposed strategy is the possibility to reduce both: the local fine mesh Degrees of Freedom (DOF) and the Lagrange multipliers DOF avoiding the space-time coupling problem. With a series of numerical examples, it was shown that POD modes may be obtained training the problem in one part of the domain and then, to use only a few of these POD modes to solve problems with the gradients in different areas without significant variations of the relative error.