A Comparison of Additive Schwarz Preconditioners for Parallel Adaptive Finite Elements

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


We consider a second order elliptic boundary value problem in the variational form: find uH01(Ω), for a given polygonal (polyhedral) domain \(\varOmega \subset \mathbb{R}^{d},\,d = 2,3\) and a source term fL2(Ω), such that
$$\displaystyle{ \underbrace{\mathop{\int _{\varOmega }\nabla u^{{\ast}}(x) \cdot \nabla v(x)\,dx}}\limits _{\equiv a(u^{{\ast}},v)} =\underbrace{\mathop{ \int _{\varOmega }f(x)v(x)\,dx}}\limits _{\equiv (f,v)},\quad \text{for all }v \in H_{0}^{1}(\varOmega ). }$$
The Bank–Holst parallel adaptive meshing paradigm [1–3] is utilised to solve (1) in a combination of domain decomposition and adaptivity.



This work was supported by the Numerical Algorithms and Intelligent Software Centre funded by the UK EPSRC grant EP/G036136 and the Scottish Funding Council.


  1. 1.
    R.E. Bank, Some variants of the Bank-Holst parallel adaptive meshing paradigm. Comput. Vis. Sci. 9(3), 133–144 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.E. Bank, M. Holst, A new paradigm for parallel adaptive meshing algorithms. SIAM J. Sci. Comput. 22(4), 1411–1443 (2000) (electronic)Google Scholar
  3. 3.
    R.E. Bank, M. Holst, A new paradigm for parallel adaptive meshing algorithms. SIAM Rev. 45(2), 291–323 (2003) (electronic). Reprinted from SIAM J. Sci. Comput. 22(4), 1411–1443 (2000) [MR1797889]Google Scholar
  4. 4.
    R.E. Bank, P.K. Jimack, S.A. Nadeem, S.V. Nepomnyaschikh, A weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial differential equations. SIAM J. Sci. Comput. 23(6), 1817–1841 (2002) (electronic)Google Scholar
  5. 5.
    M. Dryja, O.B. Widlund, Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15(3), 604–620 (1994). Iterative methods in numerical linear algebra (Copper Mountain Resort, CO, 1992)Google Scholar
  6. 6.
    S. Loisel, H. Nguyen, An optimal schwarz preconditioner for a class of parallel adaptive finite elements (submitted)Google Scholar
  7. 7.
    G. Meurant, The Lanczos and Conjugate Gradient Algorithms. Software, Environments, and Tools, vol. 19 (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006). From theory to finite precision computationsGoogle Scholar
  8. 8.
    A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsHeriot-Watt University, Riccarton, EdinburghSchwarzUK

Personalised recommendations