From: Wolfgang Bangerth Date: Wed, 14 Nov 2007 22:09:03 +0000 (+0000) Subject: More text X-Git-Tag: v8.0.0~9600 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=cfc7d7bb11781fda8c6b9f2cd1a052ac8633a760;p=dealii.git More text git-svn-id: https://svn.dealii.org/trunk@15499 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-22/doc/intro.dox b/deal.II/examples/step-22/doc/intro.dox index c8f58b01e0..c1c48d567b 100644 --- a/deal.II/examples/step-22/doc/intro.dox +++ b/deal.II/examples/step-22/doc/intro.dox @@ -180,6 +180,76 @@ advection is treated explicitly using the just-computed velocity field. +

Weak form and space discretization

+ +Like solving the mixed Laplace equations, solving the Stokes equations +requires us to choose particular pairs of finite elements for +velocities and pressure variables. In the current program, we use the +stable pair $Q_{p+1}^d \times Q_p, p\ge 1$. These are continuous +elements, so we can form the weak form of the Stokes equation without +problem by integrating by parts and substituting continuous functions +by their discrete counterparts: +@f{eqnarray*} + (\nabla {\mathbf v}_h, \eta \varepsilon ({\mathbf u}^n_h)) + - + (\nabla \cdot {\mathbf v}_h, p^n_h) + &=& + ({\mathbf v}_h, \mathrm{Ra} \; T^{n-1}_h \mathbf{g}), + \\ + (q_h, \nabla \cdot {\mathbf u}^n_h) &=& 0, +@f} +for all test functions $\mathbf v_h, q_h$. The first term of the first +equation is considered as the scalar product between tensors, i.e. +$(\nabla {\mathbf v}_h, \eta \varepsilon ({\mathbf u}^n_h))_\Omega + = \int_\Omega \sum_{i,j=1}^d [\nabla {\mathbf v}_h]_{ij} + \eta [\varepsilon ({\mathbf u}^n_h)]_{ij}$. +Because the second tensor in this product is symmetric, the +anti-symmetric component of $\nabla {\mathbf v}_h$ plays no role and +it leads to the entirely same form if we use the symmetric gradient of +$\mathbf v_h$ instead. Consequently, the formulation we consider and +that we implement is +@f{eqnarray*} + (\varepsilon({\mathbf v}_h), \eta \varepsilon ({\mathbf u}^n_h)) + - + (\nabla \cdot {\mathbf v}_h, p^n_h) + &=& + ({\mathbf v}_h, \mathrm{Ra} \; T^{n-1}_h \mathbf{g}), + \\ + (q_h, \nabla \cdot {\mathbf u}^n_h) &=& 0. +@f} + +The more interesting question is what we do with the temperature +equation. By default, not all discretizations of this equation are +equally stable unless we either do something like upwinding, +stabilization, or all of this. XXXXXXXXXX TALK ABOUT CHOICE OF FE +AND DG/CG XXXXX + +@f{eqnarray*} + (\tau_h,T^n_h) + + + k_n(\nabla \tau_h, \kappa \nabla T^n_h) + &=& + (\tau_h,T^{n-1}_h+k_n\gamma) + - + k_n (\tau_h, {\mathbf u}^n \cdot \nabla T^{n-1}_h), +@f} +for all discrete test functions $\tau_h$. One typically integrated the +last term on the right hand side by parts on each cell, yielding the +following taking into account that the velocity field is assumed to be +divergence-free: +@f{eqnarray*} + (\tau_h,T^n_h) + + + k_n(\nabla \tau_h, \kappa \nabla T^n_h) + &=& + (\tau_h,T^{n-1}_h+k_n\gamma) + + + k_n \sum_K \left\{ ({\mathbf u}^n \cdot \nabla \tau_h, T^{n-1}_h)_K - + (\tau_h, (\mathbf n \cdot {\mathbf u}^n) T^{n-1}_h)_{\partial K} \right\}. +@f} +XXXXXXXXXX WHAT DO I DO WITH THE DIV U TERM? XXXXX + +

Linear solvers