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I am studying the finite element method and have found a problem:

For a regular and quasi-uniform family of simplicial partitions, prove uniform equivalence $H^1$ norms of grid functions to the norm: $$ |u|_{1,h} = \sqrt{|u|_{L_2(\Omega)}^2 + h^{m-2} \sum\limits_{e \in \tau _h} \sum\limits_{k=1}^m(u(x_k^e)-u(x_{m+1}^e))^2}$$

The hint says about using the canonical simplex $e_0$.

I'm know that if $x = G_e y + b$ then we have $|u|^2_{L_2(e)} = |\det G_e||u|^2_{L_2(e_0)}$ and we can try do smth like : $$ |\det G_e||u|^2_{L_2(e_0)} + \sum\limits_{k=1}^m(u(x_k^e)-u(x_{m+1}^e))^2 = |\det G_e||u|^2_{L_2(e_0)} + \sum\limits_{k=1}^m(|\det G_y | \int\limits_0^{y_k}[G^{-1}_yu']_mds)^2$$ Also i have: $\det G_y = \frac{mes(e)}{mes(e_0)}$ and estimates: $\frac{\rho^m h^m}{(\sigma \nu)^{-m}} \le mes(e) \le \rho^m h^m$ and $||G^{-1}_y|| \le \frac{\sigma \nu \rho}{h}$, where $\sigma, \rho, \nu - consts$.

After that, I have no ideas. I tried to use Cauchy's inequality, but it only turns out to be an upper bound, and anyway my constants will depend on h...

Any suggestions?

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  • $\begingroup$ It looks like you are on the right tracks, this is basically all answered by scaling arguments. The paper "An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems" by E. Georgioulis shows some examples of scaling arguments with nodal evaluations. Also Brenner and Scott's book on finite elements gives a good exposure of scaling arguments. By the way, what is the index m here? $\endgroup$ Commented May 23, 2024 at 14:20
  • $\begingroup$ Do you mean the affine functions by grid functions? If so $u(y)-u(x)=Du(y-x)$ and as hint suggested calculate in $e_0$ so that you can get the harder direction easier. Then use your estimates between $e, e_0$ to conclude. $\endgroup$ Commented Jun 6, 2024 at 7:23

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