add initial abstract

Author: jeremylt

papers/high-order-bddc-2021/lfa_bddc_high_order_fem.tex

@@ -65,7 +65,13 @@
 
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 \begin{abstract}
-TODO
+Domain decomposition methods are popular preconditioners for solving linear systems derived from discretizing PDEs.
+Balancing Domain Decomposition by Constraints (BDDC) is a non-overlapping domain decomposition technique that is closely related to the dual primal version of finite element tearing and interconnecting.
+Local Fourier Analysis (LFA) is a technique for investigating and tuning preconditioners for finite difference and finite element methods.
+In this paper, we use LFAToolkit.jl, a Julia package for LFA of high-order finite element methods, to analyze BDDC for high-order finite elements.
+Specifically, we develop LFA of BDDC with arbitrary second-order PDEs using high-order finite element discretizations and examine the performance of BDDC on subdomains consisting of single and multiple high-order finite elements.
+We can replicate previous work on the LFA of BDDC on subdomains with multiple low-order elements.
+Examples are presented in two and three dimensions to validate our LFA of BDDC for the Laplacian and linear elasticity.
 \end{abstract}
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papers/references.bib

@@ -114,31 +114,6 @@ @article{fischer2020scalability
   month = {6}
 }
 
-@misc{libceed-user-manual,
-  title = {lib{C}{E}{E}{D} {U}ser {M}anual},
-  author = {
-    Abdelfattah, Ahmad and
-    Barra, Valeria and
-    Beams, Natalie and
-    Brown, Jed and
-    Camier, Jean-Sylvain and
-    Dobrev, Veselin and
-    Dudouit, Yohann and
-    Ghaffari, Leila and
-    Kolev, Tzanio and
-    Medina, David and
-    Rathnayake, Thilina and
-    Thompson, Jeremy L and
-    Tomov, Stanimire
-  },
-  month = mar,
-  year = 2021,
-  publisher = {Zenodo},
-  version = {0.8},
-  doi = {10.5281/zenodo.4895340},
-  url = {https://doi.org/10.5281/zenodo.4895340}
-}
-
 % -----------------------------------------------------------------------------
 % Multigrid
 % -----------------------------------------------------------------------------
@@ -616,6 +591,59 @@ @TechReport{petsc-user-ref
   url    = {https://www.mcs.anl.gov/petsc}
 }
 
+@misc{libceed-user-manual,
+  title = {lib{C}{E}{E}{D} {U}ser {M}anual},
+  author = {
+    Abdelfattah, Ahmad and
+    Barra, Valeria and
+    Beams, Natalie and
+    Brown, Jed and
+    Camier, Jean-Sylvain and
+    Dobrev, Veselin and
+    Dudouit, Yohann and
+    Ghaffari, Leila and
+    Kolev, Tzanio and
+    Medina, David and
+    Rathnayake, Thilina and
+    Thompson, Jeremy L and
+    Tomov, Stanimire
+  },
+  month = mar,
+  year = 2021,
+  publisher = {Zenodo},
+  version = {0.8},
+  doi = {10.5281/zenodo.4895340},
+  url = {https://doi.org/10.5281/zenodo.4895340}
+}
+
+@article{libceed,
+  title = {libCEED: Fast algebra for high-order element-based discretizations},
+  doi = {10.21105/joss.02945},
+  url = {https://doi.org/10.21105/joss.02945},
+  year = {2021},
+  publisher = {The Open Journal},
+  volume = {6},
+  number = {63},
+  pages = {2945},
+  author = {
+    Jed Brown and
+    Ahmad Abdelfattah and
+    Valeria Barra and
+    Natalie Beams and
+    Jean-Sylvain Camier and
+    Veselin Dobrev and
+    Yohann Dudouit and
+    Leila Ghaffari and
+    Tzanio Kolev and
+    David Medina and
+    Will Pazner and
+    Thilina Ratnayaka and
+    Jeremy Thompson and
+    Stan Tomov
+  },
+  journal = {Journal of Open Source Software}
+}
+
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 % Elasticity
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