Past Event: Oden Institute Seminar

Structure preserving methods for approximating fluid stresses and velocities

Jay Gopalakrishnan, Professor, Department of Mathematics & Maseeh Distinguished Chair, Portland State University

Abstract

An age-old topic of discussion in computational fluid dynamics is the proper treatment of the incompressibility constraint on the fluid velocity u, namely div(u)=0. To obtain numerical velocities that satisfy this constraint exactly, there are (at least) two categories of methods, one that requires square integrability of all derivatives of the velocity, and another that requires square integrability of only the divergence of velocity. In the latter, instead of using the standard Lagrange finite element spaces, one uses H(div)-conforming finite elements for velocity approximation. A natural question to ask in this context is what would be a natural Sobolev space for viscous fluid stresses to pair with an H(div) velocity? We report on our research into a mixed formulation with a stress space that pairs well with such spaces for velocity. The main new insight is that stresses should lie in a nonstandard Sobolev space H(curl div). We shall see that finite elements of matrix fields with continuous normal-tangential components are appropriate for approximating viscous stresses. Prized structure-preservation properties like mass conservation and pressure robustness are immediate in our newly introduced framework. Bio: Jay Gopalakrishnan is a computational mathematician whose research centers around the design of numerical methods for partial differential equations and their rapid solution by iterative techniques. He co-invented two classes of numerical methods called HDG and DPG methods and has authored over eighty publications. He currently serves in the editorial board of a journal of the Society of Industrial and Applied Mathematics and has served in the boards of six other journals. He has worked at Bell Labs, University of Minnesota, Medtronic Inc, and for over a decade, at University of Florida. In 2012 he resigned his full professorship at University of Florida to take up the Maseeh Distinguished Chair in Mathematical Sciences at Portland State University, where you will currently find him engaged in a variety of regional activities to bolster scientific computation.