Summary
We discuss some overlapping domain decomposition algorithms for solving sparse nonlinear systemof equations arising fromthe discretization of partial differential equations. All algorithms are derived using the three basic algorithms: Newton for local or global nonlinear systems, Krylov for the linear Jacobian system inside Newton, and Schwarz for linear and/or nonlinear preconditioning. The two key issues with nonlinear solvers are robustness and parallel scalability. Both issues can be addressed if a good combination of Newton, Krylov and Schwarz is selected, and the right selection is often dependent on the particular type of nonlinearity and the computing platform.
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References
Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Users Manual, ANL, 2008.
Cai, X.-C., Gropp, W., Keyes,D., Melvin, R., Young, D.P.: Parallel Newton-Kryiov-Schwarz algorithms for the transonic full potential equation, SIAM J. Sci. Comput., 19 (1998), 246–265.
Cai, X.-C., Gropp, W.D., Keyes, D.E., Tidriri, M.D.: Newton-Krylov-Schwarz methods in CFD, Proceedings of the International Workshop on the Navier-Stokes Equations, Notes in Numerical Fluid Mechanics, R. Rannacher, eds. Vieweg, Braunschweig, 1994.
Cai, X.-C., Keyes, D.E.: Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), 183–200.
Cai, X.-C., Li, X.: Inexact Newton methods with nonlinear restricted additive Schwarz preconditioning for problems with high local nonlinearities, in preparation.
Cai, X.-C., Li, X.: A domain decomposition based parallel inexact Newton’s method with subspace correction for incompressible Navier-Stokes equations, Lecture Notes in Computer Science, Springer, 2009.
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.
Eisenstat, S.C., Walker, H.F.: Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), 16–32.
Hwang, F.-N., Cai, X.-C.: A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, J. Comput. Phys., 204 (2005), 666–691.
Kelley, C.T., Keyes, D.E.: Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal., 35 (1998), 508–523.
Knoll, D., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193 (2004), 357–397.
Lanzkron, P.J., Rose, D.J., Wilkes, J.T.: An analysis of approximate nonlinear elimination, SIAM J. Sci. Comput., 17 (1996), 538–559.
Smith, B., Bjørstad, P., Gropp, W.: Domain Decomposition. Parallel Multilevel Methods for Eliptic Partial Differential Equations, Cambridge University Press, New York, 1996.
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, Oxford, 1999.
Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory, Springer, Berlin, 2005.
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Cai, XC. (2009). Nonlinear Overlapping Domain Decomposition Methods. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering, vol 70. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02677-5_23
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DOI: https://doi.org/10.1007/978-3-642-02677-5_23
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