Computational Optimization and Applications

, Volume 50, Issue 2, pp 193–221 | Cite as

The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions

  • Hannes Buchholzer
  • Christian Kanzow
  • Peter Knabner
  • Serge Kräutle


The semismooth Newton method was introduced in a paper by Qi and Sun (Math. Program. 58:353–367, 1993) and the subsequent work by Qi (Math. Oper. Res. 18:227–244, 1993). This method became the basis of many solvers for certain classes of nonlinear systems of equations defined by a nonsmooth mapping. Here we consider a particular system of equations that arises from the discretization of a reactive transport model in the subsurface including mineral precipitation-dissolution reactions. The model is highly complicated and uses a coupling of PDEs, ODEs, and algebraic equations, together with some complementarity conditions arising from the equilibrium conditions of the minerals. The aim is to show that this system, though quite complicated, usually satisfies the convergence criteria for the semismooth Newton method, and can therefore be solved by a locally quadratically convergent method. This gives a theoretical sound approach for the solution of this kind of applications, whereas the geoscientist’s community most frequently applies algorithms involving some kind of trial-and-error strategies.


Semismooth Newton method Quadratic convergence Complementarity problems Reactive transport Mineral precipitation-dissolution 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Hannes Buchholzer
    • 1
  • Christian Kanzow
    • 1
  • Peter Knabner
    • 2
  • Serge Kräutle
    • 2
  1. 1.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  2. 2.Department of MathematicsUniversity of Erlangen-NürnbergErlangenGermany

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