Abstract:
In this talk, a new concept in the simulation of partial differential equation is introduced, which allow to zoom the solution in a particular region ofinterest without increasing much cost of computation. To illustrate this concept, we consider a linear elliptic boundary value problem set in a star shapeddomain. The original domain is transformed to subdomains by homothety andit is assumed that we are interested in the solution of the problem only on oneof these subdomains, which is termed as the region of interest. To achieve thisgoal invariant embedding technique is used which furnishes operators on theboundaries of these homothetic domains (similar to the optimal feedback in thecontrol problem) and relate Dirichlet and Neuman data on the boundary. Theseoperators satisfy a Riccati equation and solving the Ricatti equation providesa boundary condition on the boundary of the region of interest such that thesolution of the related problem is exactly the restriction of the solution of theoriginal problem to the domain of interest. This problem solved on the region of interest with the same number of unknowns as the original one will yield abetter precision on thisdomain : this is the zooming effect.