By Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszynski, Waldemar Rachowicz, Adam Zdunek
Divided into sections, this publication specializes in the basics of 3-dimensional conception of HP equipment and implementation matters. It offers a number of functions that mirror numerous tasks for which the third-dimensional HP code has been used over the years.
Read or Download Computing with HP-Adaptive Finite Elements.: Volume II, Frontiers three dimensional elliptic and maxwell problems with applications PDF
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Additional resources for Computing with HP-Adaptive Finite Elements.: Volume II, Frontiers three dimensional elliptic and maxwell problems with applications
6 Pyramid Element All (master) elements discussed so far support construction of polynomial exact sequences — the corresponding spaces of shape functions consist of polynomials only. In this context, the pyramid element is essentially different, the pyramid function spaces span complete polynomials of a fixed order, but they also include nonpolynomial shape functions. Pyramid as a degenerated hexahedron. The key point behind the construction presented by Zaglmayr in  is the concept of the parametric element.
Given a distribution§ E ∈ H r −1 ( I ), we first compute its average value E 0 = E, 1 Difference E − E 0 has a zero average and, consequently, there exists a potential u ∈ H0r ( I ) such that u = E − E 0 . We now interpolate the potential u using operator ∂0 . Notice that, due to the homogeneous boundary values, the interpolation reduces to the L 2 -projection onto the element bubbles. Having determined the interpolant u2 = ∂0 u ∈ P p , we define the final interpolant of distribution E by summing up its mean with the derivative of the projection u2 −1 E = E 0 + u2 We leave for the reader to demonstrate the commutativity property.
This indicates in particular that, in context of general parametric (nonaffine) elements‡ unstructured mesh generators should be used with caution, compare . The critique does not apply to (algebraic) mesh generators based on a consistent representation of the domain as a manifold, with underlying global maps parametrizing portions of the domain. Upon a change of variables, the original problem can then be redefined in the reference domain discretized with affine elements. We will return to this issue in Chapter 5.