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Keywords:
algebraic multigrid; agglomeration; non-linear elliptic problem; nonlinear preconditioning; Newton method; finite elements
algebraic multigrid; agglomeration; non-linear elliptic problem; nonlinear preconditioning; Newton method; finite elements
Summary:
This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones and Vassilevski 2001. Then, the algebraic construction from Jones, Vassilevski and Woodward 2003 of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. A numerical illustration is also provided.
References:
[1] R. E. Bank, M. Holst: A new paradigm for parallel adaptive meshing algorithms. SIAM J. Sci. Comput. 22 (2000), 1411–1443. DOI 10.1137/S1064827599353701 | MR 1797889
[2] R. E. Bank, P. K. Jimack, S. A. Nadeem, and S. V. Nepomnyaschikh: A weakly overlapping domain decomposition preconditioner for the finite element solution of elliptic partial differential equations. SIAM J. Sci. Comput. 23 (2002), 1817–1841. DOI 10.1137/S1064827501361425 | MR 1923714
[3] P. N. Brown, P. S. Vassilevski, and C. S. Woodward: On mesh-independent convergence of an inexact Newton-multigrid algorithm. SIAM J. Sci. Comput. 25 (2003), 570–590. DOI 10.1137/S1064827502407822 | MR 2058076
[4] X.-C. Cai, D. E. Keyes: Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24 (2002), 183–200. DOI 10.1137/S106482750037620X | MR 1924420
[5] X.-C. Cai, D. E. Keyes, and L. Marcinkowski: Nonlinear additive Schwarz preconditioners and applications in computational fluid dynamics. Int. J. Numer. Methods Fluids 40 (2002), 1463–1470. DOI 10.1002/fld.404 | MR 1957600
[6] J. E. Dennis Jr., R. B. Schnabel: Numerical methods for unconstrained optimization and nonlinear equations. Classics in Applied Mathematics, Vol. 16, SIAM, Philadelphia, 1996. MR 1376139
[7] E. G. D’yakonov: Optimization in Solving Elliptic Problems. CRC Press, Boca Raton, 1996. MR 1396083
[8] J. E. Jones, P. S. Vassilevski: AMGe based on element agglomeration. SIAM J. Sci. Comput. 23 (2001), 109–133. DOI 10.1137/S1064827599361047 | MR 1860907
[9] J. E. Jones, P. S. Vassilevski, and C. S. Woodward: Nonlinear Schwarz-FAS methods for unstructured finite element problems. In: Proceedings of the Second M.I.T. Conference on Computational Fluid and Solid Mechanics, Cambridge, MA, June 17–20, 2003, Elsevier, 2003, pp. 2008–2011. MR 2029558
[10] G. Karypis, V. Kumar: METIS: A family of multilevel partitioning algorithm. http://www-users.cs.umn.edu/karypis/metis/
[11] B. Smith, P. Bjørstad, and W. Gropp: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, 1996. MR 1410757
[12] V. Thomée: Galerkin finite element methods for parabolic problems. Lecture Notes in Mathematics, Vol. 1054, Springer-Verlag, Berlin, 1984. MR 0744045
[13] P. S. Vassilevski: Sparse matrix element topology with application to AMG and preconditioning. Numer. Lin. Alg. Appl. 9 (2002), 429–444. DOI 10.1002/nla.300 | MR 1934869
[14] A. Ženíšek: Nonlinear elliptic and evolution problems and their finite element approximations. Computational Mathematics and Applications, Academic Press Inc., London, 1990. MR 1086876
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