From 482bbbb99cecbbe10105fa505d55610d8bc3fd48 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Thu, 2 Sep 2021 22:23:01 -0600 Subject: [PATCH] Minor edits to the conclusions section of step-82. --- examples/step-82/doc/results.dox | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) diff --git a/examples/step-82/doc/results.dox b/examples/step-82/doc/results.dox index 8aee1cff82..2bdee1440a 100644 --- a/examples/step-82/doc/results.dox +++ b/examples/step-82/doc/results.dox @@ -8,7 +8,7 @@ When running the program, the sparsity pattern is written to an svg file, the so Number of active cells: 64 Number of degrees of freedom: 576 Assembling the system............. -Done. +Done. DG H2 norm of the error: 0.0151063 DG H1 norm of the error: 0.000399747 L2 norm of the error: 5.33856e-05 @@ -108,6 +108,11 @@ This corresponds to the bi-Laplacian problem with the manufactured solution ment +This matches the expected optimal convergence rates for the $H^2$ and +$H^1$ norms, but is sub-optimal for the $L_2$ norm. Incidentally, this +also matches the results seen in step-47 when using polynomial degree +$k=2$. +

Possible extensions

@@ -119,18 +124,19 @@ The code can be easily adapted to deal with the following cases:
  • LDG method for the Poisson problem (use the discrete gradient consisting of the broken gradient and the lifting of the jump of $u_h$).
    • -We give below additional details for the first point. +We give below additional details for the first of these points. +

    Non-homogeneous Dirichlet boundary conditions

    If we prescribe non-homogeneous Dirichlet conditions, say @f[ -\nabla u=\mathbf{g} \quad \mbox{and} \quad u=g \qquad \mbox{on } \partial \Omega, +\nabla u=\mathbf{g} \quad \mbox{and} \quad u=g \qquad \mbox{on } \partial \Omega, @f] then the right-hand side $\boldsymbol{F}$ of the linear system needs to be modified as follows @f[ F_i:=\int_{\Omega}f\varphi_i-\sum_{e\in\mathcal{E}_h^b}\int_{\Omega}r_e(\mathbf{g}):H_h(\varphi_i)+\sum_{e\in\mathcal{E}_h^b}\int_{\Omega}b_e(g):H_h(\varphi_i)+\gamma_1\sum_{e\in\mathcal{E}_h^b}h_e^{-1}\int_e\mathbf{g}\cdot\nabla\varphi_i+\gamma_0\sum_{e\in\mathcal{E}_h^b}h_e^{-3}\int_e g\varphi_i, \qquad 1\leq i \leq N_h. @f] -Note that for any given index $i$, many of the terms are zero. Indeed, for $e\in \mathcal{E}_h^b$ we have ${\rm supp}\,(r_e(\mathbf{g}))={\rm supp}\,(b_e(g))=K$, where $K$ is the element for which $e\subset\partial K$. Therefore, the liftings $r_e(\mathbf{g})$ and $b_e(g)$ contributes to $F_i$ only if $\varphi_i$ has support on $K$ or a neighbor of $K$. In other words, when integrating on a cell $K$, we need to add +Note that for any given index $i$, many of the terms are zero. Indeed, for $e\in \mathcal{E}_h^b$ we have ${\rm supp}\,(r_e(\mathbf{g}))={\rm supp}\,(b_e(g))=K$, where $K$ is the element for which $e\subset\partial K$. Therefore, the liftings $r_e(\mathbf{g})$ and $b_e(g)$ contribute to $F_i$ only if $\varphi_i$ has support on $K$ or a neighbor of $K$. In other words, when integrating on a cell $K$, we need to add @f[ \int_{K}f\varphi_i+\sum_{e\in\mathcal{E}_h^b, e\subset\partial K}\left[-\int_{K}r_e(\mathbf{g}):H_h(\varphi_i)+\int_{K}b_e(g):H_h(\varphi_i)+\gamma_1h_e^{-1}\int_e\mathbf{g}\cdot\nabla\varphi_i+\gamma_0h_e^{-3}\int_e g\varphi_i\right] @f] -- 2.39.5