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I have a paper on this which, as long as your boundary is flat and you don't expecct it to pass vortexes, should do you well: https://doi.org/10.1002/fld.1427. The gist of it is that as long as the boundary is flat and you enforce no tangential flow, the normal traction is equal to the pressure in an incompressible flow. This is an OK BC for Poiseuille-like ...


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I am adding another answer here since this is the kind of resource I was originally looking for. The Journal of Inquiry Based Learning has a book on mathematical modelling. The abstract says the following: This is a set of notes for a one semester course in Mathematical Modeling. The topics covered are difference equations, Markov chains, Monte Carlo ...


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A great example in this vein is Lorena Barba's CFDPython also known as "12 Steps to Navier Stokes", which consists of a sequence of jupyter notebooks that go from really basic numerical analysis up through more complex problems.


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One project I have discovered since I initially asked this question is the following: https://projectlovelace.net/ It is still a work in progress, but is building up a set of problems that can aid in learning computational science. For a similar resource that focuses primarily on bioinformatics we have Rosalind: http://rosalind.info/problems/locations/


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Split $[0,1]$ into two elements $[0,1/2]$ and $[1/2,1]$. Consider the function $f$ satisfying $f(x)=0$ for $x < 1/2$ and $f(x)=1$ otherwise. For this function the $H^1$ norm is infinity but if you calculate the $H^1$ norm over the two elements separately you get 0 and $1/2$, respectively. In particular, the derivative of the Heaviside function is a delta ...


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You get a divergence-free $Z_h$ if you choose for example $V_h$ and $Q_h$ such that $\forall v_h\in V_h(div(v_h)\in Q_h)$. (An example for such a pair is the Scott-Vogelius element with $V_h=P^k,Q_h=P^{k-1}_{discont}$) Because if you take $v_h\in Z_h$ it fulfills $(div(v_h),q_h)=0 \ \forall q_h\in Q_h $. Now by the special choice of $V_h$ and $Q_h$ above we ...


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The statement as given is indeed correct (i.e., left and right hand side are different) for almost any function $f(x)$. What it says, once you write out what these norms are, is that $$ \sum_m \sqrt{\int_{T_m} f(x)^2 } \neq \sqrt{ \sum_m \int_{T_m} f(x)^2 }. $$


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I think $u - \Pi u$ is zero on a part of the boundary. Then you can use Poincare inequality to bound $\| u - \Pi u \|_0 \leq C| u - \Pi u|_1$. It could also be possible to look at the case $m = 0$ and argue the $L^2$ part is of higher order and, hence, smaller in the asymptotic limit.


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The other answer has everything you already need, but it's also worth pointing out that $u_h$ is computable whereas $I_hu$ is not: The latter requires you to know the exact solution, which in general we of course don't know (and if we did, we didn't need to compute a Galerkin approximation $u_h$).


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Let's assume $u$ is the solution to the variational problem and $u_h$ is the Galerkin approximation of the solution on the subspace $V_h \subset V$. By Cea's lemma you have: \begin{equation} ||u-u_h||_V \leq C\inf\limits_{v_h \in V_h}||u-v_h||_V \end{equation} for some positive constant $C$. Now you define the projection $I_h:V \rightarrow V_h$. As you ...


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After some digging I could finally find a paper that more or less answered my question: Bermúdez, A., Rodrıguez, R., & Santamarina, D. (2003). Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations. Journal of computational and applied mathematics, 152(1-2), 17-34. It turns out that the equation for the ...


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