313 Higher Order FE Methods for Challenging Problems in Science and Engineering

Leszek Demkowicz, ICES, University of Texas at Austin
Jay Gopalakrishnan, Portland State University
It is well known that higher order Finite Element (FE) methods offer higher convergence rates and potentially huge savings in both time and memory, provided a number of condition is met. To mention a few:

1. The anticipated solution must be sufficiently regular.
2. The particular FE method must be (discrete) stable for meshes consisting of higher order elements, potentially of different size and locally varying order of approximation to enable h-, p-, and hp-adaptivity.
3. For non-trivial, curvilinear geometries, geometry description must be accurate and efficient.
4. Mesh generation has to accommodate higher order degrees-of-freedom (dof) and be optimal, both in terms of $h$- and $p$- convergence rates. This leads to the necessity of special and efficient interpolation schemes for isoparameteric elements.
5. Data structures have to support elements of different type (e.g. the exact sequence elements: H^1-,H(curl)-,H(div)- and L^2-conforming), and different shapes (hexas, tets, prisms, pyramids).
6. Construction of shape functions should be general to accommodate elements of different shapes and conformity, and arbitrary polynomial order.
7. Integration of element matrices should use tensorization (fast integration schemes).
8. Construction of solvers and preconditioners is more flexible (e.g. p-muligrid) but also more challenging, theory- and implementation-wise.

We invite participation of all interested colleagues that pursue higher order elements in context of different methodologies including conforming and various DG methods, as well as different applications including:
- classical wave propagation and vibration problems (acoustics, Maxwell, viscoelastodynamics, coupled problems),
- non-classical wave propagation problems including modeling of metamaterials, cloaking, dispersive and nonlinear wave problems,
- classical CFD problems (compressible and incompressible Navier-Stokes), non-non-Newtonian flow, MHD etc.

While we are not limiting applications, the focus of the symposium will be on methodologies and technical implementation issues: a-posteriori error estimation and adaptivity, openMP and MPI implementations, construction of preconditioners etc.