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From: | Torquil Macdonald Sørensen |
Subject: | [Getfem-users] RT0 and Nedelec vs. Whitney 1 and 2-forms |
Date: | Thu, 23 Sep 2010 20:01:45 +0200 |
User-agent: | Mozilla/5.0 (X11; U; Linux x86_64; en-US; rv:1.9.1.12) Gecko/20100917 Icedove/3.0.8 |
Hi!I'm having trouble with calculation of mass matrix elements on a 3d tetrahedron mesh. I have two different cases. In the first case I want to use Whitney 1-form edge elements, and in the second case Whitney 2-form face elements. I'm given to understand that these are the same as the NEDELEC and RT0 elements offered by GetFEM.
The Whitney elements are defined as follows: For a point i, lambda_i is the usual affine function that is equal to 1 at i, and equal to zero at neighbouring points.
For an edge {ij} going from point i to point j, the Whitney 1-form element is lambda_ij := lambda_i grad lambda_j - lambda_j grad lambda_i The "2-form" element for a face {ijk} islambda_{ijk} := 0.5*(lambda_i (grad lambda_j) x (grad lambda_k) + cyclic permutations of {ijk}).
When varying the "fineness"/lattice spacing of the mesh for a given physical domain [0,1]^3, these definitions lead to the following behaviour of the mass matrix components:
m_ff' ~ 1/h m_ee' ~ hwhere m_ff' is a component of the mass matrix for the 2-form elements, m_ee' is the same for the 1-form elements, and h is the lattice spacing.
However, in my case, the components m_ff' decreases when h decreases. Does this indicate a difference between the definition of the RT0 element relative to the Whitney 2-form element? If so, is it possible to get one from the other by a simple multiplication by h?
Thanks Torquil
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