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I'm working in a finite element code where I need to calculate $$\int_F b_F$$ where $F$ is a face of some tetrahedron $K$ in the mesh, and $b_F$ is the usual bubble function defined by $$b_F=9\lambda_1\lambda_2\lambda_3$$ where $\lambda_i$ are the usual barycentric coordinates. Note that $\textrm{supp}(b_F)=K_+\cup K_-$, where $K_+$ and $K_-$ are the thetrahedron sharing the face $F$. Where (book o paper) can I find this value?


For example, in 2D (where $K$ is a triangle and $F$ some edge) $b_F=4\lambda_1\lambda_2$ and $$\displaystyle\int_F b_F=\frac{2}{3}\,|h_F|$$

but I need the result when $K$ is a tetrahedron.

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    $\begingroup$ My understanding of bubble functions is that they are zero on the cell faces, i.e., they have no implications for continuity across faces. In that case, the surface integral over a face or edge would of course be zero. $\endgroup$ – Wolfgang Bangerth Dec 22 '15 at 16:41
  • $\begingroup$ Thanks @WolfgangBangerth for comment. Those are the bubble functions $b_K$ defined over the element $K$, with support in $K$. But I'm using bubble functions $b_F$, whitch has its support in $K_+\cup K_-$ (where $K_+$ and $K_-$ are the element that share the face $F$) $\endgroup$ – yemino Dec 22 '15 at 19:16
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I'll assume the triangular face of your tetrahedron has straight edges since that was the case with the edges of the triangle in your 2D example.

For a straight-edged triangle, there is a simple closed-form expression for integrating a polynomial over the triangle. That expression is given, for example, in equation 15.26 of this reference: http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/IFEM.Ch15.d/IFEM.Ch15.pdf

That equation, using the notation of the listed reference is:

$$ {\frac{1}{2A}}\int_{\Omega^e}\zeta^i_1\zeta^j_2\zeta^k_3 d\Omega = {\frac{i!j!k!}{(i+j+k+2)!}}, i\ge0, j\ge0, k\ge0. $$

Instead of defining the triangular, barycentric coordinates as $\lambda_i$, this reference uses $\zeta_i$. For your function, $9\lambda_1\lambda_2\lambda_3$, i, j, and k in equation 15.26 are all one. So, by simple substitution, your integral will be equal $3A/20$ where $A$ is the area of the triangle.

By the way, there is also a closed-form expression for the integrals of a polynomial over a tetrahedron should you need to perform those. It can be found as equation 9.21 of reference: http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch09.d/AFEM.Ch09.pdf

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