journal = {Computer Methods in Applied Mechanics and Engineering}
}
+% ------------------------------------
+% Step 50
+% ------------------------------------
+
+@article{karakashian2003posteriori,
+ title={A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems},
+ author={Karakashian, Ohannes A and Pascal, Frederic},
+ journal={SIAM Journal on Numerical Analysis},
+ volume={41},
+ number={6},
+ pages={2374--2399},
+ year={2003},
+ publisher={SIAM}
+}
+
% ------------------------------------
% Step 51
% ------------------------------------
Number = {91-279},
Url = {citeseer.ist.psu.edu/saad93flexible.html}
}
+
+@Article{clevenger_par_gmg,
+ Title = {A Flexible, Parallel, Adaptive Geometric Multigrid method for FEM},
+ Author = {Thomas C. Clevenger and Timo Heister and Guido Kanschat and Martin Kronbichler},
+ Journal = {submitted},
+ Year = {2019},
+ Url = {https://arxiv.org/abs/1904.03317}
+}
This program was contributed by Thomas C Clevenger and Timo Heister.
<br>
This material is based upon work partly supported by the National
-Science Foundation Award DMS-1901529, OAC-2015848, EAR-1925575, by the Computational
+Science Foundation Award DMS-2028346, OAC-2015848, EAR-1925575, by the Computational
Infrastructure in Geodynamics initiative (CIG), through the NSF under Award
EAR-0949446 and EAR-1550901 and The University of California -- Davis.
</i>
$\epsilon = 100$ otherwise. The boundary conditions are $u=0$ on the whole boundary and
the right-hand side is $f=1$. We use continuous Q2 elements to discretize $V_h$ and use a
residual-based, cell-wise a posteriori error estimator
-$e(K) = e_{\text{cell}}(K) + e_{\text{face}}(K)$ from _CITE EST PAPER_ with
+$e(K) = e_{\text{cell}}(K) + e_{\text{face}}(K)$ from @cite karakashian2003posteriori with
@f{align*}
e_{\text{cell}}(K) = h^2 \| f + \epsilon \triangle u \|_K^2, \qquad
e_{\text{face}}(K) = \sum_F h_F \| [ \epsilon \nabla u \cdot n ] \|_F^2.
in timings we expect for GMG methods (vmults, assembly, etc.) due to the imbalance of the
mesh partition.
-_CITE MG PAPER_ contains a full discussion of the partition efficiency model
+@cite clevenger_par_gmg contains a full discussion of the partition efficiency model
and the effect the imbalance has on the GMG V-cycle timing. In summary, the value
of $\mathbb{E}$ is highly dependent on the degree of local mesh refinement used and has
an optimal value $\mathbb{E} \approx 1$ for globally refined meshes. Typically for adaptively