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Prof. Kapil Kumar Sharma
South Asian University, New Delhi, India
Computing Zoom in the Simulation of Partial Differential Equations
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.
Kapil Kumar Sharma received his post graduation in mathematics from the Kurukshetra University. He qualified National Eligibility Test for lectureship conducted by CSIR-UGC and Graduate Aptitude test in Engineering conducted by Indian Institute of Technology, India. In 2004, he has been awarded PhD in mathematics from Indian Institute of Technology Kanpur, India. His PhD dissertation established the theory for the numerical study of boundary value problems for singularly perturbed differential difference equations. In the same year he joined the Department of Mathematics (Centre for Advanced Study in Mathematics), Panjab University, Chandigarh as a faculty member. He had been awarded the prestigious INRIA post-doctoral fellowship and joined MAB-Institute de Mathematiqe, BordeuaxUniversite 1, France in 2005. There he worked with Prof. Jacques Henry, Directeur de Recherches, INRIA, on a new concept in the simulation of elliptic type partial differential equation that has been named as Computing Zoom Technique. Presently, he is working as one of the founding faculty members in the department of mathematics, South Asian University established by SAARC nations. During his more than 15 years research and teaching experience, he held many administrative positions. He guided three PhD students and three PhD students are working under his supervision. Prof. Sharma is in the editorial board of Applied Mathematics and Computation, a leading international journal in applied mathematics, published by Elsevier. Besides chairing sessions and delivering talk in different national / international conferences, he organized several symposium / workshop / seminar at national / international level. He has published around 50 research papers in refereed international journal of repute. His research interests include Singularly Perturbed Problems, Numerical Analysis of Partial Differential Equations, Finite Element Analysis and Computing Zoom Technique.
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