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Archive : Numerical Analysis and PDE

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Jie Shen, Purdue UniversityJie Shen, Purdue UniversityEWG 336Title: Efficient and accurate numerical methods for solving fractional PDEs<br><br> Abstract: We present efficient and accurate numerical methods for fractional Laplacian equations and for time-fractional diffusion equations.<br><br>For fractional Laplacian problem in bounded domains, we adopt the Caffarelli-Silvestre extension which transforms the fractional Laplacian equation in d-dimension into an equivalent system with local derivatives in (d+1)-dimension. We develop an efficient numerical method based on the generalized Laguerre approximation in the extended direction and usual (FEM or spectral) approximation in the original domain. Moreover, we enrich the spectral approximation space by using leading singular functions associated with the extended $y$-direction so that high-accuracy can be achieved despite the singularity of extended problem at $y=0$. <br><br>For time-fractional diffusion equations, we can adopt a similar approach used for the extended problem of the fractional Laplacian. However, an essential difficulty arises as the time-fractional operator is not self-adjoint which makes the diagonalization process very ill conditioned. We shall propose a novel approach to overcome this difficulty.10/18/2018 2:00:00 PM10/18/2018 3:00:00 PMFalse
Zhiqiang Cai, Purdue UniversityZhiqiang Cai, Purdue UniversityEWG 336Title: A Posteriori Error Estimation Techniques for Finite Element Methods<br> Description: Adaptive mesh renement (AMR) algorithms are one of two necessary tools for grand challenging problems in scientific computing. Reliability of computer simulations is responsible for accurate computer predictions/designs. Efficient and reliable a posteriori error estimation are, respectively, the key for success of AMR algorithms and the reliability of computer predictions/designs. Since Babuska's pioneering work in 1976, the a posteriori error estimation has been extensively studied, and impressive progress has been made during the past four decades. However, due to its extreme difficulty, this important research field of computational science and engineering remains wide open. In this talk, I will describe (1) basic principles of the a posteriori error estimation techniques for finite element approximations to partial differential equations and (2) our recent work. 10/4/2018 2:00:00 PM10/4/2018 3:00:00 PMFalse
Fengyan Li, Rensselaer Polytecnic InstituteFengyan Li, Rensselaer Polytecnic InstituteEWG 336Title: Asymptotic preserving discontinuous Galerkin methods for some kinetic models in a diffusive scaling<br> Abstrct: Asymptotic preserving (AP) methods are numerical schemes that are designed to work uniformly with respect to different scales or regimes of a model as its parameter varies, while mimicking the asymptotic limit. In this talk, we are concerned with AP methods based on discontinuous Galerkin (DG) discretizations for some kinetic models in a diffusive scaling. We will first examine and review the AP property of the upwind DG methods, then report on our developments in the design and analysis of AP methods. These methods are based on reformulations of the equations and involve high order DG methods in space as well as suitably selected temporal discretizations<br> NOTE: This Seminar is scheduled for a different day and time than usual.9/28/2018 3:00:00 PM9/28/2018 4:00:00 PMFalse
Yangwen Zhang, University of DelawareYangwen Zhang, University of DelawareEWG 336Title: An HDG method for Dirichlet Boundary Control of PDEs <br><br> Abstract: We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems for PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. In this talk, we use an HDG method for a Dirichlet boundary control problem for PDEs. We obtain well-posedness and regularity results for the Dirichlet control problem, and we prove optimal a priori error estimates in 2D for the control in both the high regularity and low regularity cases. Moreover, we present numerical experiments to demonstrate the performance of the HDG methods and illustrate our numerical analysis results. 9/20/2018 2:00:00 PM9/20/2018 3:00:00 PMFalse

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