Commuting shape functions for $ H(\operatorname {div})$ and $ H(\operatorname {curl})$

Electromagnetic field simulations are based on the vector valued function spaces $ H(\operatorname {div})$ and $ H(\operatorname {curl})$. Many properties of the finite element discretization (such as interpolation operators, preconditioners, a posteriori estimates etc) are tightly connected to the commuting diagram:

\begin{displaymath}\begin{array}{ccccccc} H^1 & \stackrel{\nabla}{\longrightarro... ...atorname{div}}{\longrightarrow} & S_h \:. \\ [8pt] \end{array}\end{displaymath} (1)

In short, the diagram shows the relation of the function spaces in the first line, and corresponding relations of the finite element spaces in the second line. S. Zaglmayr developed high order finite elements for $ H(\operatorname {div})$ and $ H(\operatorname {curl})$, and implemented the 2D case. These new shape functions are commuting in the following sense: The gradients of $ H^1$ edge shape functions are $ H(\operatorname {curl})$-edge-shape functions. The same hold for face shape functions, and also, the $ \operatorname{curl}$ of $ H(\operatorname {curl})$ face shape functions are $ H(\operatorname {div})$ face shape functions. The advantage of the new construction is that simple preconditioners work well. Some of these shape functions are drawn in Figure 5.

\includegraphics[width=6cm]{curvedshaft}    \includegraphics[width=6cm]{curvedshaft}
Figure 5: Edge shape function ($ p=5$) and inner shape function ($ p=8$)

The $ H(\operatorname {curl})$ finite elements have been used for a 2D computation of a C - magnet, see Figure 6. A coil around the limb (on the left hand side) drives the magnetic flux in the magnet. The flux spreads out in the air gap on the right hand side. The simulation used elements of order $ 8$.

\includegraphics[width=10cm]{curvedshaft}
Figure 6: Magnetic flux in the C-magnet

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