Download Computing with HP-Adaptive Finite Elements.: Volume II, by Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej PDF

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.

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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

Sample text

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 [175] 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 [18]. 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.

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