From 812e06e294de908186b50d823e3c13435f0a6e95 Mon Sep 17 00:00:00 2001 From: Timo Heister Date: Tue, 14 May 2019 11:43:28 -0600 Subject: [PATCH] small fixes --- examples/step-63/doc/intro.dox | 10 +++++----- examples/step-63/step-63.cc | 23 +++++++++++------------ 2 files changed, 16 insertions(+), 17 deletions(-) diff --git a/examples/step-63/doc/intro.dox b/examples/step-63/doc/intro.dox index 61f5df839c..f12dcad9a9 100644 --- a/examples/step-63/doc/intro.dox +++ b/examples/step-63/doc/intro.dox @@ -35,7 +35,7 @@ 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. Define the \textit{Peclet number}: $\mathcal{P}\coloneqq +3. Define the 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 @@ -68,11 +68,11 @@ 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) +\boldsymbol{\beta}\cdot \nabla u_h-f,\,\delta_K +\boldsymbol{\beta}\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 +where the sum is done over all cells $K$ with the inner product taken +for each cell, and $\delta_K$ is a cell-wise constant stabilization parameter defined in On Discontinuity-Capturing Methods for Convection-Diffusion Equations diff --git a/examples/step-63/step-63.cc b/examples/step-63/step-63.cc index 0e061b3f2c..4efa00baad 100644 --- a/examples/step-63/step-63.cc +++ b/examples/step-63/step-63.cc @@ -339,17 +339,15 @@ namespace Step63 // @sect3{Right-hand Side and Boundary Values} - // The problem solved in this tutorial is an adaptation of Ex. 3.1.3 - // found on pg. 118 of - // // Finite Elements and Fast Iterative Solvers: with Applications in - // Incompressible Fluid Dynamics by Elman, Silvester, and Wathen with - // Applications in Incompressible Fluid Dynamics by Elman, Silvester, and - // Wathen. The main difference being that we add a hole in the center of our - // domain with zero Dirichlet boundary. + // Incompressible Fluid Dynamics by Elman, Silvester, and Wathen. The + // main difference being that we add a hole in the center of our domain with + // zero Dirichlet boundary. - // We have a zero right-hand side. + // We have a zero right-hand side: template class RightHandSide : public Function { @@ -448,10 +446,11 @@ namespace Step63 // @sect3{Streamline Diffusion} - // Streamline diffusion stabilization term. Value is defined in - // - // On Discontinuity—Capturing Methods for Convection—Diffusion Equations - // by Volker and Petr + // Streamline diffusion stabilization constant. Parameter design is taken + // from On + // Discontinuity-Capturing Methods for Convection-Diffusion Equations by + // Volker John and Petr Knobloch template double compute_stabilization_delta(const double hk, const double eps, -- 2.39.5