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Archive : Numerical Analysis and PDE

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Joshua Padgett, Texas Tech UniversityJoshua Padgett, Texas Tech UniversityEWG 336Title: Operator splitting methods for approximating singular nonlinear differential equations <br><br> Abstract: Operator splitting techniques were originally introduced in an effort to save computational costs in numerical simulations. Classically, such methods were restricted to dimensional splitting of evolution operators. However, these methods have since been extended to allow for splitting of problems involving nonlinear operators which evolve on vastly different time scales. In this talk I will introduce the notion of nonlinear operator splitting and rigorously justify the approach by considering some techniques from Lie group theory. This is a nonstandard presentation that should also be accessible to graduate students. The second half of the talk will provide results concerning two very interesting applications of operator splitting techniques: nonlinear stochastic problems and singular combustion problems. The former problems have traditionally been plagued with low-order techniques with restrictive regularity conditions, while the latter have the need for strongly adaptive methods which recover important qualitative properties. We will discuss in detail how operator splitting provides solutions to these issues, while also being straightforward to implement.11/29/2018 3:00:00 PM11/29/2018 4:00:00 PMFalse
Bernardo Cockburn, University of MinnesotaBernardo Cockburn, University of MinnesotaEWG336Event on <b> Tuesday</b> 11/13</br> Title: Variational principles for discontinuous Galerkin methods: A short story. <br><br> Abstract: By the late 60's, the most successful finite element methods for solid mechanics had been obtained by using the principle minimal potential energy and the principle of complementary energy. We show how each of these minimization principles gives rise to four different types of finite element methods, namely, the original minimization problem, its mixed form, its hybrid form, and its hybridized mixed form. Using this framework, we then show how, after a slow evolution lasting more than four decades, the hybrid methods associated to the principle of minimal potential energy and the hybridized mixed methods associated to the principle of minimal complementary energy converged to a single method, the so-called hybridizable discontinuous Galerkin method, by two apparently different ways related to the stabilization procedures distinctive of the original discontinuous Galerkin method. 11/13/2018 3:00:00 PM11/13/2018 4:00:00 PMFalse
Ke Shi, Old Dominion UniversityKe Shi, Old Dominion UniversityEWG 336Title: Mixed DG method and HDG method for stationary incompressible magnetohydrodynamics <br><br> Description: Magnetohydrodynamics (MHD) models have various important applications in liquid metal industry, controlled fusion and astronomy etc. There have been extensive discussions on numerical methods for MHD models. However, due to the complicated nonlinear coupling and rich structures of MHD systems, the numerical simulation still remains a challenging and active research area. In this talk We introduce a mixed DG method and a HDG method for stationary magnetohydrodynamics (MHD) systems. In all cases, the schemes provide optimal convergence for the primary unknowns under minimal regularity assumptions for the exact solution. 11/8/2018 3:00:00 PM11/8/2018 4:00:00 PMFalse
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

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