Computing with adaptive higher-order finite elements

Event Detail

Event type: Applied Mathematics and Computation Seminar
Date/Time: 11/02/2009-12:00   
Location: GILK 113

Speaker info

Speaker: Pavel Solin, University of Nevada, Reno

Abstract:
In this talk I will briefly introduce our group, its motivation and goals, some results that we obtained so far, and what we hope to accomplish in the future. In particular, I will explain why we believe that it makes sense to design numerical methods that are independent of the partial differential equations (PDE) solved. The current standard in my field is to develop sophisticated error estimates and techniques that only work (or work efficiently) for a narrow class of PDE. In contrast to that, our methods work in the same way for any PDE or multiphysics PDE system, and they can even be used in other fields such as computer graphics or robotics. Examples will be shown. Main ideas of adaptive higher-order finite element methods (hp-FEM) will be explained and their extremely fast, exponential convergence will be demonstrated via live computations and compared to the slow, algebraic convergence of traditional low-order FEM. I will mention the main ideas of our novel adaptive multimesh hp-FEM that makes it possible to discretize a PDE system monolithically (i.e., as if it was a single equation) while using individual (optimal) meshes for all physical fields in the system. Numerical comparisons to standard (single-mesh) hp-FEM will be presented. We will show that even fluid flows can be handled as multiphysics problems with velocity components and pressure approximated using different meshes. We will show that the multimesh hp-FEM makes it possible to design simple and efficient adaptive algorithms for time-dependent PDE problems based on dynamically-changing meshes. (Automatic adaptivity for time-dependent problems is much more difficult than adaptivity for statonary ones because of the necessity of mesh unrefinement). I will explain the main idea and show several computational videos related to various problems (flame propagation, thermally-conductive flow, microwave heating, thermoelasticity, single and two-phase flow, heat and moisture transfer in concrete, Gross-Pitaevski equation of quantum physics, etc). If some time is left, I will say a few words about our open source project FEMhub whose objective is to create an open source distribution of finite element codes with unified Python interface.