From: scott.miller Date: Mon, 26 Aug 2013 02:18:39 +0000 (+0000) Subject: more intro for step-51 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4a2f89cfeeea7cf2bb431aa5be8f4d2f289ae17c;p=dealii-svn.git more intro for step-51 git-svn-id: https://svn.dealii.org/trunk@30491 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-51/doc/intro.dox b/deal.II/examples/step-51/doc/intro.dox index 22dc984e08..1c14fa37c9 100644 --- a/deal.II/examples/step-51/doc/intro.dox +++ b/deal.II/examples/step-51/doc/intro.dox @@ -37,14 +37,14 @@ The Dirichlet-to-Neumann map concept then permits the solution procedure:
  1. Solve a Dirichlet problem on each element individually, where the local/interior element solutions no longer couple to neighboring elements. This is known as the -'local solver', and is an embarrissingly parallel element-by-element +'local solver', and is an embarrassingly parallel element-by-element solution process. -
  2. Use the local element solutions to construct the global problem for the +
  3. Use the local element equations to construct the global problem for the trace values. These are the only globally coupled unknowns. -
  4. Reconstruct the local solutions from the computed trace values. +
  5. Solve for the element-local solutions using the computed trace values.
-The above procedure also has a linear algebra interpretation and referred to +The above procedure also has a linear algebra interpretation and is referred to as static condensation. Let us write the complete linear system associated to the HDG problem as a block system with the discrete DG variables U as first block and the skeleton variables $\Lambda$ as the second block: @@ -73,7 +73,7 @@ degrees of freedom of the interior DG variables which are always only related to a single cell. The coupling to other cells is introduced by the matrices B and C over the skeleton variable. The block-diagonality of A and the structure in B and C allow us to invert the -matrix A element by element (the local solution of the Dirichelt +matrix A element by element (the local solution of the Dirichlet problem) and subtract $CA^{-1}B$ from $D$.

Solution quality and rates of convergence

@@ -103,7 +103,7 @@ We consider the convection-diffusion equation: \quad \text{ in } \Omega, \\ u &=& g_D, \quad \text{ on } \partial \Omega_D, \\ (\mathbf{c} u - \kappa \nabla u)\cdot \mathbf{n} &=& g_N, - \quad \text{ on } \partial \Omega_D. + \quad \text{ on } \partial \Omega_N. @f} Introduce the auxiliary variable $\mathbf{q}=-\kappa \nabla u$ and rewrite @@ -113,7 +113,62 @@ the above equation as the first order system: \nabla \cdot (\mathbf{c} u + \mathbf{q}) &=& f, \quad \text{ in } \Omega, \\ u &=& g_D, \quad \text{ on } \partial \Omega_D, \\ (\mathbf{q} + \mathbf{c}u)\cdot\mathbf{n} &=& g_N, - \quad \text{ on } \partial \Omega_D. + \quad \text{ on } \partial \Omega_N. +@f} + +We multiply these equations by the weight functions $\mathbf{v}, w$ +and integrate by parts over every element $K$ to obtain: +@f{eqnarray*} + (\kappa^{-1} \mathbf{q}, \mathbf{v})_K - (u, \nabla\cdot\mathbf{v})_K + + <\hat{u}, \mathbf{v}\cdot\mathbf{n}>_{\partial K} &=& 0, \\ + - (\mathbf{c} u + \mathbf{q}, \nabla w)_K + + <(\hat{\mathbf{c} u}+\hat{\mathbf{q}})\cdot\mathbf{n}, w>_{\partial K} + &=& (f,w)_K. +@f} + +The terms decorated with a $\hat{}$ denote the numerical traces (also commonly referred +to as numerical fluxes). They are approximations +to the interior values on the boundary of the element. To ensure conservation, +these terms must be single-valued on any given element edge $\partial K$. +We eliminate the numerical trace $\hat{\mathbf{q}}$ by using traces of the form: +@f{eqnarray*} + \hat{\mathbf{c} u}+\hat{\mathbf{q}} = \mathbf{c}\hat{u} + \mathbf{q} + + \tau(u - \hat{u})\mathbf{n} \quad \text{ on } \partial K. +@f} +The local stabilization parameter $\tau$ has effects on stability and accuracy of +HDG solutions; see the literature for a further discussion. + +The trace/skeleton variables in HDG methods are single-valued on element faces. As such, +they must strongly represent the Dirichlet data on $\partial\Omega_D$. We introduce +a new variable $\lambda$ such that +@f{equation*} + \hat{u} = + \begin{cases} + g_D & \text{ on } \partial \Omega_D, \\ + \lambda & \text{otherwise}. +\end{cases} +@f} + +Eliminating $\hat{u}$ from the weak form in favor of $\lambda$, summing the elemental +contributions across all elements in the triangulation, and enforcing the normal +component of the numerical flux, we arrive at the final form of the problem: +Find $(\mathbf{q}, u, \lambda)$ such that +@f{eqnarray*} + (\kappa^{-1} \mathbf{q}, \mathbf{v})_{\mathcal{T}} + - (u, \nabla\cdot\mathbf{v})_{\mathcal{T}} + + <\lambda, \mathbf{v}\cdot\mathbf{n}>_{\partial\mathcal{T}} + &=& + - _{\partial\Omega_D}, +\\ + - (\mathbf{c} u + \mathbf{q}, \nabla w)_{\mathcal{T}} + + <(\hat{\mathbf{c}u} + \hat{\mathbf{q}})\cdot\mathbf{n}, w>_{\partial \mathcal{T}} + &=& + (f, w)_{\mathcal{T}}, +\\ + < \left[ \negthinspace \left[ (\hat{\mathbf{c}u} + \hat{\mathbf{q}})\cdot\mathbf{n} + \right] \negthinspace \right], \mu>_{\partial \mathcal{T}} + &=& + _{\partial\Omega_N} @f}