Many numerical simulations require accurate visualization through the use of e.g. streamlines, streaklines and isosurfaces. This generally requires knowledge of both the physics behind what is being modelled and the numerics of how the approximation is generated. This can be quite challenging for higher-order numerical methods as many visualization techniques assume low order approximations with high degree of continuity. This PhD project will focus on using the mathematics behind the approximation to create better visualization techniques. Specifically, it will focus on the challenges that come with geometry of irregularly shaped domains as well as provable error characteristics. The main part of this project focuses on the analysis and computation of finite element methods, specifically the discontinuous Galerkin method. The PhD will have the opportunity to work for extended periods at the University of Utah (www.sci.utah.edu). This is a U.S. Air Force Office of Scientific Research (AFOSR) project and is in collaboration with Mike Kirby (School of Computing and SCI Institute, University of Utah).

(i) Michael Steffen, Sean Curtis, Robert M. Kirby, and Jennifer K. Ryan. "Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinuous fields." IEEE Transactions on Visualization and Computer Graphics,
14(3):680-692, 2008.
(ii) David Walfisch, Jennifer K. Ryan, Robert M. Kirby, and Robert Haimes. "One-sided
smoothness-increasing accuracy-conserving filtering for enhanced streamline integration
through discontinuous fields." Journal of Scientific Computing, 38(2):164-184, 2009.
(iii) Hanieh Mirzaee, Liangyue Ji, Jennifer K. Ryan, and Robert M. Kirby. "Smoothness-increasing accuracy-conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes."  SIAM Journal on Numerical Analysis, 49:1899–1920, 2011.