Selected aspects of hp-FEM in 3D

Pavel Kus, Pavel Solin, Tomas Vejchodsky

Finite element method is one of the most often used methods for solving partial
differential equations. The ability of adaptive refinement of the mesh is
a substantial feature of all modern methods. It has been shown theoretically
and practically, that for a variety of physical and technical problems, the
ability to change both size and polynomial order of elements is very usefull.
Such variant of the finite element method is called hp-FEM.

The main drawback of this method is its difficult implementation, which is even
more significant in three spatial dimensions. In order to ensure conformity of
the approximation, we have to care about orientation of faces and edges when 
constructing base functions. The most demanding part is the treatment of hanging
nodes, that appear during adaptation steps. According to our previous experience
from 2D, we decided to implement arbitrary level hanging nodes. This implementation
is more complicated, but it significantly simplifies the adaptive procedure (no 
forced refinements are needed) and leads to smaller discrete systems (degrees of 
freedom are used only when they are really needed).

In this presentation, we discuss some aspects of the implementation of the method.
Numerical example is presented to show its usefullness.