Extension of hp-Interface.

diff --git a/9.2/paper.bib b/9.2/paper.bib
index c135102e3537399fd7542ac68ef4699beaa7823b..1b1d05f3c3960ef1352c98dd4fb554e0340aab76 100644 (file)
--- a/9.2/paper.bib
+++ b/9.2/paper.bib
@@ -950,3 +950,76 @@ pages = {171--233},
 year = 1982,
 doi = {10.1016/0096-3003(82)90191-6}
 }
+
+@article{BabuskaSuri1990,
+  title = {The p- and h-p versions of the finite element method, an overview},
+  author = {Babuška, I. and Suri, M.},
+  year = {1990},
+  journal = {Computer Methods in Applied Mechanics and Engineering},
+  volume = {80},
+  pages = {5--26},
+  issn = {0045-7825},
+  doi = {10.1016/0045-7825(90)90011-A},
+  number = {1}
+}
+
+@article{Mavriplis1994,
+  title = {Adaptive mesh strategies for the spectral element method},
+  author = {Mavriplis, C.},
+  year = {1994},
+  journal = {Computer Methods in Applied Mechanics and Engineering},
+  volume = {116},
+  pages = {77--86},
+  issn = {0045-7825},
+  doi = {10.1016/S0045-7825(94)80010-3},
+  number = {1}
+}
+
+@article{MelenkWohlmuth2001,
+  title = {On residual-based a posteriori error estimation in \textit{hp}-{{FEM}}},
+  author = {Melenk, J. M. and Wohlmuth, B. I.},
+  year = {2001},
+  journal = {Advances in Computational Mathematics},
+  volume = {15},
+  pages = {311--331},
+  issn = {1019-7168, 1572-9044},
+  doi = {10.1023/A:1014268310921},
+  number = {1-4}
+}
+
+@inproceedings{HoustonSeniorSueli2003,
+  title = {Sobolev regularity estimation for hp-adaptive finite element methods},
+  booktitle = {Numerical {{Mathematics}} and {{Advanced Applications}}},
+  author = {Houston, P. and Senior, B. and Süli, E.},
+  editor = {Brezzi, Franco and Buffa, Annalisa and Corsaro, Stefania and Murli, Almerico},
+  year = {2003},
+  pages = {631--656},
+  publisher = {{Springer Milan}},
+  location = {{Milano}},
+  doi = {10.1007/978-88-470-2089-4_58},
+  isbn = {978-88-470-2089-4}
+}
+
+@article{HoustonSueli2005,
+  title = {A note on the design of \textit{hp}-adaptive finite element methods for elliptic partial differential equations},
+  author = {Houston, P. and Süli, E.},
+  year = {2005},
+  journal = {Computer Methods in Applied Mechanics and Engineering},
+  volume = {194},
+  pages = {229--243},
+  issn = {0045-7825},
+  doi = {10.1016/j.cma.2004.04.009},
+  number = {2}
+}
+
+@article{EibnerMelenk2007,
+  title = {An adaptive strategy for \textit{hp}-{{FEM}} based on testing for analyticity},
+  author = {Eibner, T. and Melenk, J. M.},
+  year = {2007},
+  journal = {Computational Mechanics},
+  volume = {39},
+  pages = {575--595},
+  issn = {1432-0924},
+  doi = {10.1007/s00466-006-0107-0},
+  number = {5}
+}

diff --git a/9.2/paper.tex b/9.2/paper.tex
index bb3aee5fe37f99ac4673c6eee2604c4b805f34b5..59ce0bf6d8a4adee1c38686fc5a1f959ad975592 100644 (file)
--- a/9.2/paper.tex
+++ b/9.2/paper.tex
@@ -446,7 +446,61 @@ meshes with more than 4B unknowns. {\color{red}TODO[Timo]}
 \subsection{Better support for parallel $hp$-adaptive algorithms}
 \label{subsec:hp}
 
-\todo[inline]{Marc: Your section}
+With the previous release, \dealii supports $hp$-adaptive finite element methods
+on distributed memory systems \cite{dealII91}. We implemented the bare functionality
+for $hp$-adaptive methods with the objective to offer the greatest flexibility in
+their application. Here, reference finite elements still had to be assigned manually
+to each cell, which may not lead to an optimal mesh and is tedious.
+
+With the current release, $hp$-adaptive finite element methods have been further
+expanded in both the range of features in terms of decision strategies and their 
+user interface, effectively making the more attractive to use. We introduced many
+new functions that automatize the general workflow for applying $hp$-decision
+strategies, which run on top of the previous low-level implementation for both serial
+and parallel applications.
+
+The interface is now as simple to use as the one for pure adaptive mesh refinement.
+Consider the following (incomplete) listing as an example: We estimate both error and
+smoothness of the finite element approximation. Further, we flag certain fractions of
+cells with the highest and lowest errors for refinement and coarsening, respectively
+(here: 30\% / 3\%). From those cells listed for adaptation, we specify a subset of them
+for $h$- and $p$-adaptation (here: 10\% / 90\%).
+\begin{c++}
+Vector<float> estimated_error_per_cell (n_active_cells);
+KellyErrorEstimator::estimate(
+  dof_handler, ..., solution, estimated_error_per_cell, ...);
+GridRefinement::refine_and_coarsen_fixed_number(
+  triangulation, estimated_error_per_cell, 0.3, 0.03);
+
+Vector<float> estimated_smoothness_per_cell (n_active_cells);
+SmoothnessEstimator::Legendre::coefficient_decay(
+  ..., dof_handler, solution, estimated_smoothness_per_cell);
+hp::Refinement::p_adaptivity_fixed_number(
+  dof_handler, estimated_smoothness_per_cell, 0.9, 0.9);
+hp::Refinement::choose_p_over_h (dof_handler);
+
+triangulation.execute_coarsening_and_refinement();
+\end{c++}
+
+In particular, we implemented decision strategies based on refinement history and
+smoothness estimation, and made sure that they work for refinement as well as
+coarsening in terms of $h$- and $p$-adaptation in serial and parallel applications.
+
+The former relies on knowing an estimate for the upper error bound \cite[Thm.~3.4]{BabuskaSuri1990}.
+For successive refinements, we can predict how the error will change on basis of
+current error estimates and adaptation flags. In the next refinement cycle, these
+predicted error estimates allow us to decide whether the choice of adaptation in
+the previous cycle was justified, and provide the foundation for it in the next
+one \cite{MelenkWohlmuth2001}.
+
+In general, $p$-refinement is favorable over $h$-refinement in smooth regions of
+the finite element approximation \cite[Thm.~3.4]{BabuskaSuri1990}. Thus, estimating
+its smoothness provides a suitable decision indicator for $hp$-adaptation. For this
+purpose, we expand the finite element approximation into a series of orthogonal
+basis functions of increasing complexity, and consider the decay of their expansion
+coefficients as the estimation of smoothness. This has been implemented for both
+Fourier coefficients \cite{BangerthKayserHerold2007} and Legendre coefficients
+\cite{Mavriplis1994,HoustonSeniorSueli2003,HoustonSueli2005,EibnerMelenk2007}.
 
 
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