This project studies algorithms for numerical approximation of complex systems of unsteady nonlinear partial differential equations arising in underground porous media problems. It aims at designing novel inexact algebraic and linearization solvers, with each step being adaptively steered, interconnecting at any time all parts of the numerical simulation algorithm. Its key ingredient are optimal a posteriori estimates on the error in the approximate solution which give a guaranteed global upper bound, guaranteed local lower bounds, robustness with respect to the problem parameters, and which distinguish the different error components like the spatial, temporal, regularization, linearization, and algebraic solver ones. Computer implementation, assessment on academic and industrial benchmarks, and applications to contemporary environmental problems like underground storage of dangerous waste or geological sequestration of CO2 are envisaged.


  • novel multilevel algebraic solvers tailored to porous media simulations
  • novel adaptive inexact Newton linearizations
  • mass balance on each algorithm step
  • full adaptivity with local stopping criteria
  • theoretical proofs of guaranteed global reliability, local efficiency, convergence, and optimality
  • interconnection of modeling, analysis of PDEs, numerical analysis, numerical linear algebra, and scientific computing

Final goals

  • total simulation error is certified at any time
  • important reduction of the current computational burden

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