From 2a95faf0ddeb3153297dd1057a299a7ef2977e2d Mon Sep 17 00:00:00 2001 From: bangerth Date: Tue, 27 Sep 2011 03:08:38 +0000 Subject: [PATCH] Discuss the discretization issue. git-svn-id: https://svn.dealii.org/trunk@24446 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-32/doc/intro.dox | 69 +++++++++++++++++++++++++- 1 file changed, 67 insertions(+), 2 deletions(-) diff --git a/deal.II/examples/step-32/doc/intro.dox b/deal.II/examples/step-32/doc/intro.dox index 6c22a7cc1b..bdd4871880 100644 --- a/deal.II/examples/step-32/doc/intro.dox +++ b/deal.II/examples/step-32/doc/intro.dox @@ -398,8 +398,6 @@ We use the discrete form of the right hand side of this as our approximation $\tilde S$ to $S$. -@todo discretization with FE_DGP -

Changes to the artificial viscosity stabilization

@todo Martin, can you take another look at this section? In particular, I @@ -502,6 +500,73 @@ $\alpha=2$ instead, and in that case one has to re-tune the parameter. It turns out that $c_R=0.5$ works with $\alpha=2$. +

Locally conservative Stokes discretization

+ +The standard Taylor-Hood discretization for Stokes, using the $Q_{k+1}^d +\times Q_k$ element, is globally conservative, i.e. $\int_{\partial\Omega} +\mathbf n \cdot \mathbf u_h = 0$. This can easily be seen: the weak form of +the divergence equation reads $(q_h, \textrm{div}\; \mathbf u_h)=0, \forall +q_h\in Q_h$. Because the pressure space does contain the function $q_h=1$, we +get +@f{align*} + 0 = (1, \textrm{div}\; \mathbf u_h)_\Omega + = \int_\Omega \textrm{div}\; \mathbf u_h + = \int_{\partial\Omega} \mathbf n \cdot \mathbf u_h +@f} +by the divergence theorem. This property is important: if we want to use the +velocity field $u_h$ to transport along other quantities (such as the +temperature in the current equations, but it could also be concentrations of +chemical substances or entirely artificial tracer quantities) then the +conservation property guarantees that the amount of the quantity advected +remains constant. + +That said, there are applications where this global property is not +enough. Rather, we would like that it holds locally, on every +cell. This can be achieved by using the space +$Q_{k+1}^d \times DGP_k$ for discretization, where we have replaced the +continuous space of tensor product polynomials of degree $k$ for the +pressure by the discontinuous space of the complete polynomials of the +same degree. (Note that tensor product polynomials in 2d contain the functions +$1, x, y, xy$, whereas the complete polynomials only have the functions $1,x,y$.) +This space turns out to be stable for the Stokes equation. + +Because the space is discontinuous, we can now in particular choose the test +function $q_h(\mathbf x)=\chi_K(\mathbf x)$, i.e. the characteristic function +of cell $K$. We then get in a similar fashion as above +@f{align*} + 0 + = (q_h, \textrm{div}\; \mathbf u_h)_\Omega + = (1, \textrm{div}\; \mathbf u_h)_K + = \int_K \textrm{div}\; \mathbf u_h + = \int_{\partial K} \mathbf n \cdot \mathbf u_h, +@f} +showing the conservation property for cell $K$. This clearly holds for each +cell individually. + +There are good reasons to use this discretization. As mentioned above, this +element guarantees conservation of advected quantities on each cell +individually. A second advantage is that the pressure mass matrix we use as a +preconditioner in place of the Schur complement becomes block diagonal and +consequently very easy to invert. However, there are also downsides. For one, +there are now more pressure variables, increasing the overall size of the +problem, although this doesn't seem to cause much harm in practice. More +importantly, though, the fact that now the divergence integrated over each +cell is zero when it wasn't before does not guarantee that the divergence is +pointwise smaller. In fact, as one can easily verify, the $L_2$ norm of the +divergence is larger for this than for the standard Taylor-Hood +discretization. (However, both converge at the same rate to zero, since it is +easy to see that +$\|\textrm{div}\; u_h\|= +\|\textrm{div}\; (u-u_h)\|= +\|\textrm{trace}\; \nabla (u-u_h)\|\le +\|\nabla (u-u_h)\|={\cal O}(h^{k+2})$.) It is therefore not a priori clear +that the error is indeed smaller just because we now have more degrees of +freedom. + +Given these considerations, it remains unclear which discretization one should +prefer. Consequently, we leave that up to the user and make it a parameter in +the input file which one to use. +

Parallelization on clusters

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