From: Diane Guignard Date: Sat, 11 Sep 2021 22:09:12 +0000 (-0400) Subject: Minor updates in intro and results X-Git-Tag: v9.4.0-rc1~962^2~5 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=dc30f3a3f4697576a1d23c631b40713f89ea7d73;p=dealii.git Minor updates in intro and results --- diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib index 435dc4bc89..423863bfcc 100644 --- a/doc/doxygen/references.bib +++ b/doc/doxygen/references.bib @@ -1204,30 +1204,50 @@ % ------------------------------------ @article{Pryer2014, -author = {T. Pryer}, -title = {Discontinuous {G}alerkin methods for the p-biharmonic equation from a discrete variational perspective}, -journal = {Electronic Transactions of Numerical Analysis}, -volume = {41}, -pages = {328--349}, -year = {2014} + author = {T. Pryer}, + title = {Discontinuous {G}alerkin methods for the p-biharmonic equation from a discrete variational perspective}, + journal = {Electronic Transactions of Numerical Analysis}, + volume = {41}, + pages = {328--349}, + year = {2014} } @article{Bonito2021, -author = {A. Bonito and R.H. Nochetto and D. Ntogkas}, -title = {{DG} Approach to Large Bending Plate Deformations with Isometry Constraint}, -journal = {Mathematical Models and Methods in Applied Sciences}, -volume = {31}, -number = {1}, -pages = {133--175}, -year = {2021} + author = {A. Bonito and R.H. Nochetto and D. Ntogkas}, + title = {{DG} Approach to Large Bending Plate Deformations with Isometry Constraint}, + journal = {Mathematical Models and Methods in Applied Sciences}, + volume = {31}, + number = {1}, + pages = {133--175}, + year = {2021} } @book{DiPietro2011, -author = {D.A. Di Pietro and A. Ern}, -title = {Mathematical Aspects of Discontinuous {G}alerkin Methods}, -series = {Math{\'e}matiques et Applications}, -publisher={Springer Berlin Heidelberg}, -year={2011}, + author = {D.A. Di Pietro and A. Ern}, + title = {Mathematical Aspects of Discontinuous {G}alerkin Methods}, + series = {Math{\'e}matiques et Applications}, + publisher={Springer Berlin Heidelberg}, + year={2011}, +} + +@misc{BGNY2020, + title={{LDG} approximation of large deformations of prestrained plates}, + author={A. Bonito and D. Guignard and R.H. Nochetto and S. Yang}, + journal={submitted}, + year={2020}, + eprint={2011.01086}, + archivePrefix={arXiv}, + primaryClass={math.NA} +} + +@misc{BGNY2021, + title={Numerical analysis of the LDG method for large deformations of prestrained plates}, + author={A. Bonito and D. Guignard and R.H. Nochetto and S. Yang}, + journal={submitted}, + year={2021}, + eprint={2106.13877}, + archivePrefix={arXiv}, + primaryClass={math.NA} } % ------------------------------------ diff --git a/examples/step-82/doc/intro.dox b/examples/step-82/doc/intro.dox index ae81fe7f00..a84de5e0ca 100644 --- a/examples/step-82/doc/intro.dox +++ b/examples/step-82/doc/intro.dox @@ -16,7 +16,7 @@ In this example, we consider the local discontinuous Galerkin (LDG) metho \nabla u & = \mathbf{0} \quad \mbox{on } \partial\Omega, \\ u & = 0 \quad \mbox{on } \partial\Omega, @f} -where $\Omega\subset\mathbb{R}^d$ $(d=2,3)$ is an open bounded Lipschitz domain and $f\in L^2(\Omega)$. This is the same problem we have already considered in step-47, but we will here take a different approach towards solving it: Rather than using continuous finite elements and enforcing the necessary continuity via penalty terms, we will use lifting operators to deal with the fact that the usual finite element spaces are not twice continuous. +where $\Omega\subset\mathbb{R}^d$ $(d=2,3)$ is an open bounded Lipschitz domain and $f\in L^2(\Omega)$. This is the same problem we have already considered in step-47, but we will take here a different approach towards solving it: Rather than using continuous finite elements and the interior penalty method, we consider discontinuous finite elements and the local discontinous Galerkin method defined using lifting operators. The weak formulation of this problem reads as follows: find $u\in H_0^2(\Omega)$ such that @f[ @@ -129,9 +129,9 @@ The sparsity pattern associated with the above LDG method is slightly larger tha @f[ \|v_h\|_{H_h^2(\Omega)}^2\dealcoloneq\|D_h^2v_h\|_{L^2(\Omega)}^2+\sum_{e\in\mathcal{E}_h}h_e^{-1}\|\jump{\nabla_h v_h}\|_{L^2(e)}^2+\sum_{e\in\mathcal{E}_h}h_e^{-3}\|\jump{v_h}\|_{L^2(e)}^2 @f] -for any choice of penalty parameters $\gamma_0,\gamma_1>0$. In other words, the stability of the method is ensured for any positive parameters. This is in contrast with interior penalty methods for which they need to be large enough. (See also the discussions about penalty parameters in the step-39, step-47, and step-74 programs about penalty parameters.) +for any choice of penalty parameters $\gamma_0,\gamma_1>0$. In other words, the stability of the method is ensured for any positive parameters. This is in contrast with interior penalty methods for which they need to be large enough. (See also the discussions about penalty parameters in the step-39, step-47, and step-74 programs.)
  • If $\{v_h\}_{h>0}\subset \mathbb{V}_h$ is a sequence uniformly bounded in the $\|\cdot\|_{H_h^2(\Omega)}$ norm such that $v_h\rightarrow v$ in $L^2(\Omega)$ as $h\rightarrow 0$ for some $v\in H^2(\Omega)$, then the discrete Hessian $H_h(v_h)$ weakly converges to $D^2v$ in $[L^2(\Omega)]^{2\times 2}$ as $h\rightarrow 0$. Note that the uniform boundedness assumption implies that the limit $v$ belongs to $H_0^2(\Omega)$.
    • -
  • The use of a reconstructed operator simplifies the design of the numerical algorithm. In particular, no integration by parts is needed to derive the discrete problem. This strategy of replacing differential operators by appropriate discrete counter-parts can be applied to nonlinear and more general problems, for instance variational problems without a readily accessible strong formulation. +
  • The use of a reconstructed operator simplifies the design of the numerical algorithm. In particular, no integration by parts is needed to derive the discrete problem. This strategy of replacing differential operators by appropriate discrete counter-parts can be applied to nonlinear and more general problems, for instance variational problems without a readily accessible strong formulation. It has been used for instance in @cite BGNY2020 and @cite BGNY2021 in the context of large bending deformation of plates.
    • diff --git a/examples/step-82/doc/results.dox b/examples/step-82/doc/results.dox index 9fc5b85289..0437327e8f 100644 --- a/examples/step-82/doc/results.dox +++ b/examples/step-82/doc/results.dox @@ -95,17 +95,6 @@ This corresponds to the bi-Laplacian problem with the manufactured solution ment 1.277e-06 1.91 - - 7 - 16384 - 147456 - 8.870e-04 - 1.01 - 2.001e-06 - 1.97 - 3.298e-07 - 1.95 - This matches the expected optimal convergence rates for the $H^2$ and