+%-------------------------------------------------------------------------------
+% Step 1
+%-------------------------------------------------------------------------------
+
+@article{Mu05,
+ author = {Mo Mu},
+ title = {{PDE.M}art: A Network-based Problem-solving Environment for {PDE}s},
+ journal = {ACM Trans. Math. Software.},
+ year = 2005,
+ volume = 31,
+ number = 4,
+ pages = {508--531}
+}
+
+%-------------------------------------------------------------------------------
+% Step 6
+%-------------------------------------------------------------------------------
+
+@article{Kel74,
+ author = {R. Bruce Kellogg},
+ title = {On the {P}oisson equation with intersecting interfaces},
+ journal = {Applicable Analysis},
+ year = 1974,
+ volume = 4,
+ number = 2,
+ pages = {101--129}
+}
+
+%-------------------------------------------------------------------------------
+% Step 14
+%-------------------------------------------------------------------------------
+
@inproceedings{Ban00w,
author = {Wolfgang Bangerth},
title = {Mesh Adaptivity and Error Control for a Finite Element
booktitle = {Proceedings of the Fifth International Conference on Mathematical
and Numerical Aspects of Wave Propagation (Waves2000), Santiago
de Compostela, Spain, 2000},
- pages = {725--729},
year = 2000,
editor = {Alfredo Berm\'udez and Dolores G\'omez and Christophe Hazard and
Patrick Joly and Jean E. Roberts},
- publisher = {SIAM}
+ publisher = {SIAM},
+ pages = {725--729}
}
@phdthesis{Ban02,
type = {Dissertation},
year = 1996
}
+
+%-------------------------------------------------------------------------------
+% Step 18
+%-------------------------------------------------------------------------------
+
+@article{CTZ04,
+ author = {Stéphane Commend and Andrzej Truty and Thomas Zimmermann},
+ title = {Stabilized finite elements applied to elastoplasticity: I. Mixed displacement–pressure formulation},
+ journal = {Computer Methods in Applied Mechanics and Engineering},
+ year = 2004,
+ volume = 193,
+ number = 33,
+ pages = {3559--3586}
+}
(While the second mesh is entirely artificial and made-up, and
certainly not very practical in applications, to everyone's surprise it
-has found its way into the literature: see the paper by M. Mu
-titled "PDE.MART: A network-based problem-solving environment", ACM
-Trans. Math. Software, vol. 31, pp. 508-531, 2005. Apparently it is
+has found its way into the literature: see @cite Mu05. Apparently it is
good for some things at least.)
book by Simo and Hughes on “Computational Inelasticity” for a
comprehensive overview of computational strategies for solving plastic
models. Alternatively, a brief but concise description of an algorithm for
-plasticity is given in an article by S. Commend, A. Truty, and Th. Zimmermann,
-titled “Stabilized finite elements applied to
-elastoplasticity: I. Mixed displacement-pressure formulation”
-(Computer Methods in Applied Mechanics and Engineering, vol. 193,
-pp. 3559-3586, 2004).
+plasticity is given in @cite CTZ04.
<h5>Stabilization issues</h5>
the solution to not be in $W^{1,\infty}$ any more.
The typical example one uses for this is called the <i>Kellogg problem</i>
-(referring to the paper "On the Poisson equation with intersecting interfaces"
-by R. B. Kellogg, Applicable Analysis, vol. 4, pp. 101-129, 1974), which
-in the commonly used form has a coefficient $a(\mathbf x)$ that has different values
-in the four quadrants of the plane (or eight different values in the octants
-of ${\mathbb R}^3$). The exact degree of regularity (the $s$ in the
-index of the Sobolev space above) depends on the values of $a(\mathbf x)$ coming
-together at the origin, and by choosing the jumps large enough, the
-regularity of the solution can be made as close as desired to $H^1$.
+(referring to @cite Kel74), which in the commonly used form has a coefficient
+$a(\mathbf x)$ that has different values in the four quadrants of the plane
+(or eight different values in the octants of ${\mathbb R}^3$). The exact degree
+of regularity (the $s$ in the index of the Sobolev space above) depends on the
+values of $a(\mathbf x)$ coming together at the origin, and by choosing the
+jumps large enough, the regularity of the solution can be made as close as
+desired to $H^1$.
To implement something like this, one could replace the coefficient
function by the following (shown here only for the 2d case):