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On Nonlinear Dirichlet—Neumann Algorithms for Jumping Nonlinearities

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Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 55)

Abstract

We consider a quasilinear elliptic transmission problem where the nonlinearity changes discontinuously across two subdomains. By a reformulation of the problem via a Kirchho. transformation, we first obtain linear problems on the subdomains together with nonlinear transmission conditions and then a nonlinear Steklov– Poincaré interface equation. We introduce a Dirichlet–Neumann iteration for this problem and prove convergence to a unique solution in one space dimension. Finally we present numerical results in two space dimensions suggesting that the algorithm can be applied successfully in more general cases.

Keywords

  • Hydraulic Conductivity
  • Space Dimension
  • Domain Decomposition
  • Domain Decomposition Method
  • Richards Equation

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  • DOI: 10.1007/978-3-540-34469-8_61
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References

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  2. H. Berninger, Domain Decomposition Methods for Problems with Jumping Nonlinearities, PhD thesis, Freie Universität Berlin, 2006. In preparation.

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  3. R. Kornhuber, On constrained Newton linearization and multigrid for variational inequalities, Numer. Math., 91 (2002), pp. 699–721.

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  4. A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999.

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  5. M. T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44 (1980), pp. 892–898. 496

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© 2007 Springer

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Berninger, H., Kornhuber, R., Sander, O. (2007). On Nonlinear Dirichlet—Neumann Algorithms for Jumping Nonlinearities. In: Widlund, O.B., Keyes, D.E. (eds) Domain Decomposition Methods in Science and Engineering XVI. Lecture Notes in Computational Science and Engineering, vol 55. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34469-8_61

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