With all of the interest in high-order methods, it is still not clear whether these methods 

have an advantage over their low-order counterparts for practical problems. In this talk, 

high-order hp-finite element moving mesh formulations for problems with moving boundaries and discontinuous solutions will be discussed. In particular, the attainable order of accuracy will be examined for several practical applications. These applications include ducted wind turbines where the rotor is modeled with an actuator disc, solidification of silicon with a moving free-surface and solidification front, and re-entry vehicles with a bow shock where the bow shock is tracked by the moving mesh. In all of these cases the solutions have non-smooth behaviors which makes obtaining design order of accuracy difficult / impossible. We examine whether high-order accurate schemes are still advantageous for these types of problems for both uniform mesh refinement approaches and adaptive meshing. We then propose a coordinate transform that allows high-order methods to obtain optimal accuracy even for problems with singularities. The coordinate transform has two additional benefits. First, it allows controllable aspect ratio adaptive boundary layer meshing, and second, even for singular problems, high-order convergence can be obtained with a nested mesh refinement. This last fact allows Richardson extrapolation to be used to obtain high-order a-posteriori error estimates in quantities of interest.

Bio:

Prof. Brian Helenbrook currently holds the Paynter-Krigman Endowed Chair in Engineering Science Simulation at Clarkson University. He obtained his B.S. degree from the 

University of Notre Dame and a Master’s and Ph.D. degree from Princeton University