5 votes
Accepted

Quintic Hermite shape functions

I mentioned how to do it for cubic polynomials in a previous answer. And has an expanded version in my blog. You can do the derivation in global coordinates and obtain the following global basis ...
nicoguaro's user avatar
  • 8,490
4 votes
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Chebyshev/Lagrange polynomials in spectral methods

Your understanding is perfectly fine, except for the last statement that Lagrange polynomials turn out to be a more suitable choice. In fact, both methods, the modal and the nodal expansion, have ...
davidhigh's user avatar
  • 3,127
3 votes

Chebyshev/Lagrange polynomials in spectral methods

To complete david's answer: References: Canuto et al., Spectral Methods Fundamentals in Single Domains We consider the Burgers equation \begin{align} \text{advective form}: \qquad \frac{\partial u}{\...
ConvexHull's user avatar
  • 1,290
2 votes
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Nodal functionals in finite element analysis

If you have functions $\phi_1,\ldots,\phi_n : \mathbb{R}\to \mathbb{R}$ they satisfy the Lagrange property on $x_1,\ldots,x_n\in\mathbb{R}$ if $\phi_j(x_i) = \delta_{ij}$. This allows linear ...
lightxbulb's user avatar
  • 1,994
2 votes

Automatic differentiation of barycentric rational functions

$ \def\a{\alpha} \def\b{\beta} \def\e{\gamma} \def\f{\delta} \def\o{{\tt1}} \def\c{\cdot} \def\d{\oslash} \def\m{\odot} \def\qiq{\quad\implies\quad} \def\g#1#2{\frac{d #1}{d #2}} \def\LR#1{\left(#1\...
greg's user avatar
  • 594
2 votes

Conservative interpolation from a 1D grid to another 1D grid

I'll add some thoughts and terminology. First, your question doesn't make that much sense or is a bit underspecified. You are given a function values on a grid $\{x_i\}$, and you want the ...
davidhigh's user avatar
  • 3,127
1 vote

Conservative interpolation from a 1D grid to another 1D grid

You can't do this with interpolation. Imagine for example that you are representing the function $f(x)=x^2$ on the interval $[-1,1]$ on a mesh $X_{old}$ with a very small mesh width. This ...
Wolfgang Bangerth's user avatar

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