From: tcclevenger Date: Tue, 14 May 2019 17:15:59 +0000 (-0600) Subject: fix right test function and Peclet def X-Git-Tag: v9.1.0-rc2~14^2~1 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=11e1c28eef32f9cfaed330a4b064f09989e769b5;p=dealii.git fix right test function and Peclet def --- diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox index d18a9fc56a..61f5df839c 100644 --- a/examples/step-63/doc/intro.dox +++ b/examples/step-63/doc/intro.dox @@ -35,9 +35,11 @@ direction, and $f$ is a source. A few notes: 2. If $\varepsilon=0$ then this is the stationary advection equation solved in step-9. -3. If $\varepsilon \ll \|\boldsymbol{\beta}\|$, we say the problem is -advection-dominated, else we say the problem is -diffusion-dominated. +3. Define the \textit{Peclet number}: $\mathcal{P}\coloneqq +\|\boldsymbol{\beta}\| \cdot L/\varepsilon$. If $\mathcal{P}>1$, we say the +problem is advection-dominated, else if $\mathcal{P}<1$ we will +say the problem is diffusion-dominated. Here $L$ is the length +scale of the domain. For the discussion in this tutorial we will be concerned with advection-dominated flow. @@ -49,33 +51,36 @@ a(u_h,v_h) = F(v_h) @f} where @f{align*}{ -a(u_h,v_h) &= (\varepsilon \nabla u_h,\, \nabla v_h) + (\boldsymbol{\beta}\cdot\nabla u_h,\,v_h),\\ -F(v_h) &= (f,\,v_h). +a(u_h,v_h) &= (\varepsilon \nabla v_h,\, \nabla u_h) + +(v_h,\,\boldsymbol{\beta}\cdot \nabla u_h),\\ +F(v_h) &= (v_h,\,f). @f}

Streamline diffusion

The following error estimate can be shown for this PDE: @f{align*}{ -\|\nabla (u-u_h)\| \leq (1+\mathcal{P}) \inf_{v_h} \|\nabla (u-v_h)\| +\|\nabla (u-u_h)\| \leq (1+\mathcal{P}) \inf_{v_h} \|\nabla (u-v_h)\|. @f} -where $\mathcal{P} \sim \|\boldsymbol{\beta}\|/\varepsilon$ (referred -to as the Peclet number). This implies that we may have poor -numerical solutions when $\varepsilon \ll \|\boldsymbol{\beta}\|$. To -combat this, we will consider the new weak form + +Given the definition of $\mathcal{P}$ above, we may have poor +numerical solutions when $\varepsilon \ll \|\boldsymbol{\beta}\|\cdot +L$. To combat this, we will consider the new weak form @f{align*}{ -a(u_h,\,v_h) + \sum_K (-\varepsilon \Delta u_h + \boldsymbol{\beta}\cdot -\nabla u_h-f,\,\delta \boldsymbol{\beta}_K\cdot \nabla v_h)_K = F(v_h) +a(u_h,\,v_h) + \sum_K (-\varepsilon \Delta u_h + +\boldsymbol{\beta}\cdot \nabla u_h-f,\,\delta +\boldsymbol{\beta}_K\cdot \nabla v_h)_K = F(v_h) @f} -where the sum is done over all cells $K$ with the inner product is understood -on that cell, and $\delta_K$ is a cell-wise constant stabilization parameter -defined in On -Discontinuity-Capturing Methods for Convection-Diffusion Equations by Volker -John and Petr Knobloch. Essentially, adding in the discrete strong form -residual enhances the coercivity of the bilinear form $a(\cdot,\cdot)$ which -increases the stability of the discrete solution. This method is commonly -referred to as streamline diffusion or SUPG (streamline -upwind/Petrov-Galerkin). +where the sum is done over all cells $K$ with the inner product is +understood on that cell, and $\delta_K$ is a cell-wise constant +stabilization parameter defined in +On Discontinuity-Capturing Methods for Convection-Diffusion Equations +by Volker John and Petr Knobloch. Essentially, adding in the +discrete strong form residual enhances the coercivity of the bilinear +form $a(\cdot,\cdot)$ which increases the stability of the discrete +solution. This method is commonly referred to as streamline +diffusion or SUPG (streamline upwind/Petrov-Galerkin).

Smoothers

diff --git a/examples/step-63/doc/results.dox b/examples/step-63/doc/results.dox index fe3193f73b..76ca966d88 100644 --- a/examples/step-63/doc/results.dox +++ b/examples/step-63/doc/results.dox @@ -20,8 +20,8 @@ The point-wise smoothers ("Jacobi" and "SOR") get applied in the order the DoFs are numbered on each level. We can influence this using the DoFRenumbering namespace. The block smoothers are applied based on the ordering we set in @p setup_smoother. We can visualize this numbering. The -following picture shows the cell numbering of the active cells in upstream, -random, and downstream numbering (left to right): +following picture shows the cell numbering of the active cells in downstream, +random, and upstream numbering (left to right):