From 31c566e7750b53d1407cb017dc6dac0bc9243277 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Mon, 26 Mar 2018 14:36:51 -0600 Subject: [PATCH] Minor edits to grammar and formulas. --- Distributed_LDG_Method/README.md | 21 +++++++++++++-------- 1 file changed, 13 insertions(+), 8 deletions(-) diff --git a/Distributed_LDG_Method/README.md b/Distributed_LDG_Method/README.md index 27eff22..b56c6d8 100644 --- a/Distributed_LDG_Method/README.md +++ b/Distributed_LDG_Method/README.md @@ -18,7 +18,7 @@ to build discontinuous Galerkin (DG) methods. While this is very convenient, I could not use this framework for solving my research problem and I -needed write the LDG method from scratch. I thought it +needed to write the LDG method from scratch. I thought it would be helpful for others to have access to this example that goes through writing a discontinuous Galerkin method from scatch and also shows how to do it in a distributed setting using the @@ -32,7 +32,7 @@ is very useful when one is working with a differential equation and desires both approximations to the scalar unknown function as well as its flux. The application of a mixed method offers a mechanism whereby one can obtain both the scalar unknown function as well as its flux, however, -the LDG method has less degrees of freedom compared to the +the LDG method has fewer degrees of freedom compared to the mixed method with the Raviart-Thomas element. It also approximates the scalar unknown function and its flux using discontinuous polynomial basis functions and are much more @@ -95,17 +95,22 @@ w^{+} (\textbf{x} ) \, \vert_{\partial \Omega_{e} } \; = \; We define the average and jump of a function across an element face as, @f{align} -\{f\} \; = \; \frac{1}{2}(f^-+f^+), +\{f\} \; = \; \frac{1}{2}(f^-+f^+) , \qquad \mbox{and} \qquad -\left[ f \right] \; = \; f^+ \textbf{n}^+ + f^- \textbf{n}^-, +\left[ f \right] +\; = \; f^+ \textbf{n}^+ + f^- \textbf{n}^- +\; = \; (f^+ - f^-) \textbf{n}^+, @f} and, @f{align} \{\textbf{f} \} \; = \; \frac{1}{2}(\textbf{f}^- + \textbf{f}^+), \qquad \mbox{and}\qquad -\left[ \textbf{f} \right] \; = \;\textbf{f}^+ \cdot \textbf{n}^+ + -\textbf{f}^- \cdot \textbf{n}^- , +\left[ \textbf{f} \right] +\; = \; +\textbf{f}^+ \cdot \textbf{n}^+ + \textbf{f}^- \cdot \textbf{n}^- +\; = \; +(\textbf{f}^+ - \textbf{f}^-) \cdot \textbf{n}^+ , @f} where $f$ is a scalar function and $\textbf{f}$ is vector-valued function. @@ -117,7 +122,7 @@ We note that for faces that are on the boundary of the domain we have, @f} -We denote the volume integrals and surface integrals using the $L^{2}(\Omega)$ +We denote the volume integrals and surface integrals using the $L^{2}$ inner products by $( \, \cdot \, , \, \cdot \, )_{\Omega}$ and $\langle \, \cdot \, , \, \cdot \, \rangle_{\partial \Omega}$ respectively. @@ -390,4 +395,4 @@ Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation -# The Commented Code \ No newline at end of file +# The Commented Code -- 2.39.5