# Tag Info

### Finding Shape Functions for a Triangle in 3D coordinate space

This is actually quite simple. Let's say you have your (two-dimensional) reference triangle $\hat K=\left\{(\xi,\eta)\in {\mathbb R}^2: 0\le \xi \le 1, 0\le \eta\le 1, \xi \le (1-\eta)\right\}$. Then ...
• 56.1k
Accepted

### Measuring the extent to which two sets of vectors span the same space

The classical tool for this job is canonical angles. The canonical angles between $\operatorname{Im} A$ and $\operatorname{Im} B$ can be computed as $\arccos \sigma_i$, where $\sigma_i$ are the ...
• 11.8k
Accepted

### Definition of Lagrange nodes in Gmsh

I think you've got slightly the wrong end of the stick from the documentation. As with a lot of other software in the area, GMSH started out with low order, hard coded numberings. These are the ones ...
• 2,269

• 11.8k
1 vote

### How to perform FFT from plane-wave basis function coefficients to real space?

The whole point of the FFT is to avoid the costly matrix-vector multiplication of the DFT, which has an effort of $\mathcal O(N^2)$ (where $N = N_x N_y N_z$), and perform the unitary transformation ...
• 3,197
1 vote

### Finding Shape Functions for a Triangle in 3D coordinate space

From @Wolfgang Bangerth's answer, there exists a mapping function $(x,y,z)=\Phi(\xi,\eta)$ which can be expressed as a function of the basis shape functions. Using a similar notation as @Wolfgang ...
• 172
1 vote

### Computing size of N-Dimensional Polynomial Basis and Efficient Representation of Basis

As for how to create a linear index for the polynomial terms, let's consider an arrangement of terms that works nice for deduction. The terms for each dimension are enumerated as $a,b,c,\dots$. A one ...
• 171
1 vote

### Using SVD to biorthogonalize left and right eigenvectors?

I guess, the final result is suffering from instability (relative) of Gram-Schmidt, even in modified/stabilized forms. In this case, instead of going from the start: modifying the orthogonalization ...
• 8,702

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