From: Wolfgang Bangerth Date: Mon, 18 Jul 2022 22:49:20 +0000 (-0600) Subject: Fix formula mark-up. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=ebb93a20d53fedaab7966eb2b49a5d2eccd4a9d0;p=code-gallery.git Fix formula mark-up. --- diff --git a/advection_reaction_estimator/README.md b/advection_reaction_estimator/README.md index 65558e5..3787940 100644 --- a/advection_reaction_estimator/README.md +++ b/advection_reaction_estimator/README.md @@ -23,10 +23,19 @@ If you run `./DG_advection_reaction parameters.prm`, an error message will tell ## The problem: This program solves the problem, for $\Omega \in \mathbb{R^2}$ -$$\begin{cases} b \cdot \nabla u + c u = f \qquad \text{in } \Omega \\ -\qquad \qquad u=g \qquad \text{on } \partial_{-}\Omega \end{cases}$$ +@f[ +\begin{cases} b \cdot \nabla u + c u = f \qquad \text{in } \Omega \\ +\qquad \qquad u=g \qquad \text{on } \partial_{-}\Omega \end{cases} +@f] -where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in \partial \Omega: b(x)\cdot n(x) <0\}$ is the inflow part of the boundary, with $b=(b_1,b_2) \in \mathbb{R^2}$. As we know from classical DG theory, we need to ensure that $$c(x) - \frac{1}{2}\nabla \cdot b \geq \gamma_0 >0$$for some positive $\gamma_0$ so that we have coercivity in $L^2$ at the continuous level. Discrete coercivity is achieved by using a stronger norm which takes care of jumps, see Di-Pietro Ern [1] for details. +where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in +\partial \Omega: b(x)\cdot n(x) <0\}$ is the inflow part of the +boundary, with $b=(b_1,b_2) \in \mathbb{R^2}$. As we know from +classical DG theory, we need to ensure that +@f[ +c(x) - \frac{1}{2}\nabla \cdot b \geq \gamma_0 >0 +@f] +for some positive $\gamma_0$ so that we have coercivity in $L^2$ at the continuous level. Discrete coercivity is achieved by using a stronger norm which takes care of jumps, see Di-Pietro Ern [1] for details. ## The weak formulation: @@ -35,37 +44,52 @@ where $g \in L^2(\partial_{-}\Omega)$ and $\partial_{-}\Omega=\{ x \in \partial As trial space we choose $V_h = \{ v_h \in L^2(\Omega): v_h \in P^1(\mathbb{T_h})\} \notin H^1(\Omega)$. If we integrate by parts and sum over all cells -$$\sum_{T \in \mathbb{T}_h} \Bigl( (-u,\beta \cdot \nabla v_h) _T + (c u,v_h)_T + \bigl<(b \cdot n) u ,v_h \bigr>_{\partial T} \Bigr) = (f,v_h)_{\Omega}$$ +@f[ +\sum_{T \in \mathbb{T}_h} \Bigl( (-u,\beta \cdot \nabla v_h) _T + (c +u,v_h)_T + \bigl<(b \cdot n) u ,v_h \bigr>_{\partial T} \Bigr) = +(f,v_h)_{\Omega} +@f] and use the so-called DG magic formula and exploit the property $[bu]_{\mathbb{F}^i} = 0$ where $\mathbb{F}^i$ are set of internal faces we obtain the (unstable!) formulation: Find $u_h \in V_h$: -$$ +@f[ a_h(u_h,v_h) + b_h(u_h,v_h)=l(v_h) \qquad \forall v_h \in V_h -$$ +@f] where -$$ +@f[ a_h(u,v_h)=\sum_{T \in \mathbb{T}_h} \Bigl( (-u,b \cdot \nabla v_h) _T + (c u,v_h)_T \Bigr) -$$ +@f] -$$ b_h(u,v_h)= \sum_{F \not \in \partial_{-}\Omega} \bigl< \{ b u\}, [v_h]\bigr>_F $$ +@f[ +b_h(u,v_h)= \sum_{F \not \in \partial_{-}\Omega} \bigl< \{ bu\}, [v_h]\bigr>_F +@f] -$$ +@f[ l(v_h)= (f,v_h)_{\Omega} - \sum_{F \in \partial_{-}\Omega} \bigl< (b \cdot n) g,v_h \bigr>_F -$$ +@f] It's well known this formulation is coercive only in $L^2$, hence the formulation is unstable as we don't "see" the derivatives. To stabilize this, we can use a jump-penalty term, i.e. our $b_h$ is replaced by: -$$b_h^s(u_h,v_h)=b_h(u_h,v_h)+ \sum_{F \in \mathbb{F}^i} \bigl< c_F [u_h],[v_h] \bigr> $$ +@f[ +b_h^s(u_h,v_h)=b_h(u_h,v_h)+ \sum_{F \in \mathbb{F}^i} \bigl< c_F +[u_h],[v_h] \bigr> +@f] where $c_F>0$ is a function on each edge such that $c_F \geq \theta |b \cdot n|$ for some positive $\theta$. In this program, $\theta=\frac{1}{2}$ and $c_F = \frac{1}{2} |b \cdot n|$, which corresponds to an upwind formulation. Notice that consistency is trivially achieved, as $[u]_{\mathbb{F}^i} =0$. This formulation is stable in the energy norm -$$ |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2 \Bigr)^{\frac{1}{2}}$$ +@f[ + |||\cdot ||| = \Bigl(||\cdot||_{0,\Omega}^2 + \sum_{F \in + \mathbb{F}}||c_F^{\frac{1}{2}}[\cdot] ||_{0,F}^2 + \Bigr)^{\frac{1}{2}} +@f] (well defined on $H^1(\Omega) + V_h$) and moreover we have the a-priori bound: -$$|||u-u_h||| \leq C h^{k+\frac{1}{2}}||u||_{k+1,\Omega} $$ +@f[ +|||u-u_h||| \leq C h^{k+\frac{1}{2}}||u||_{k+1,\Omega} +@f] valid for $u \in H^{k+1}(\Omega)$. @@ -79,7 +103,11 @@ The estimator is the one proposed by Georgoulis, Edward Hall and Charalambos Mak The reliability is: -$$|||u-u_h|||^2 \leq C || \sqrt{b \cdot n}[u_h]||_{\Gamma^{-}}^2 + C \sum_{T \in \mathbb{T}_h}\Bigl( ||\beta (g-u_h^+)||_{\partial_{-}T \cap \partial_{-} \Omega}^2 +||f-c u_h - \Pi(f- cu_h)||_T^2 \Bigr)$$ +@f[ +|||u-u_h|||^2 \leq C || \sqrt{b \cdot n}[u_h]||_{\Gamma^{-}}^2 + C +\sum_{T \in \mathbb{T}_h}\Bigl( ||\beta (g-u_h^+)||_{\partial_{-}T +\cap \partial_{-} \Omega}^2 +||f-c u_h - \Pi(f- cu_h)||_T^2 \Bigr) +@f] where: @@ -119,4 +147,4 @@ If we look at the decrease of the energy norm of the error in the globally refin * [1] Emmanuil H. Georgoulis, Edward Hall and Charalambos Makridakis (2013), Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems. DOI: [10.1007/978-3-319-01818-8_8 ](https://link.springer.com/chapter/10.1007%2F978-3-319-01818-8_8) * [2] Di Pietro, Daniele Antonio and Ern, Alexandre (2012), Mathematical Aspects of Discontinuous Galerkin Methods. ISBN: [978-3-642-22980-0](https://www.springer.com/gp/book/9783642229794) -* [3] Franco Brezzi, Luisa Donatella Marini and Endre Süli (2004) Discontinuous Galerkin Methods for First-Order Hyperbolic Problems. DOI: [10.1142/S0218202504003866](https://doi.org/10.1142/S0218202504003866) \ No newline at end of file +* [3] Franco Brezzi, Luisa Donatella Marini and Endre Süli (2004) Discontinuous Galerkin Methods for First-Order Hyperbolic Problems. DOI: [10.1142/S0218202504003866](https://doi.org/10.1142/S0218202504003866)