Polyhedral Methods in Geosciences

1. Front Matter

   Pages i-xiii

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2. Non-conforming Finite Elements on Polytopal Meshes

   • Jérôme Droniou, Robert Eymard, Thierry Gallouët, Raphaèle Herbin

   Pages 1-35

3. Error Estimates for the Gradient Discretisation Method on Degenerate Parabolic Equations of Porous Medium Type

   • Clément Cancès, Jérôme Droniou, Cindy Guichard, Gianmarco Manzini, Manuela Bastidas Olivares, Iuliu Sorin Pop

   Pages 37-72

4. Nodal Discretization of Two-Phase Discrete Fracture Matrix Models

   • Konstantin Brenner, Julian Hennicker, Roland Masson

   Pages 73-118

5. An Introduction to Multi-point Flux (MPFA) and Stress (MPSA) Finite Volume Methods for Thermo-poroelasticity

   • Jan Martin Nordbotten, Eirik Keilegavlen

   Pages 119-158

6. High–order Discontinuous Galerkin Methods on Polyhedral Grids for Geophysical Applications: Seismic Wave Propagation and Fractured Reservoir Simulations

   • Paola F. Antonietti, Chiara Facciolà, Paul Houston, Ilario Mazzieri, Giorgio Pennesi, Marco Verani

   Pages 159-225

7. A Hybrid High-Order Method for Multiple-Network Poroelasticity

   • Lorenzo Botti, Michele Botti, Daniele A. Di Pietro

   Pages 227-258

8. The Mixed Virtual Element Method for the Richards Equation

   • Dibyendu Adak, Gianmarco Manzini, Sundararajan Natarajan

   Pages 259-297

9. Performances of the Mixed Virtual Element Method on Complex Grids for Underground Flow

   • Alessio Fumagalli, Anna Scotti, Luca Formaggia

   Pages 299-329

10. Back Matter

    Pages 331-332

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The last few years have witnessed a surge in the development and usage of discretization methods supporting general meshes in geoscience applications. The need for general polyhedral meshes in this context can arise in several situations, including the modelling of petroleum reservoirs and basins, CO₂ and nuclear storage sites, etc. In the above and other situations, classical discretization methods are either not viable or require ad hoc modifications that add to the implementation complexity. Discretization methods able to operate on polyhedral meshes and possibly delivering arbitrary-order approximations constitute in this context a veritable technological jump. The goal of this monograph is to establish a state-of-the-art reference on polyhedral methods for geoscience applications by gathering contributions from top-level research groups working on this topic. This book is addressed to graduate students and researchers wishing to deepen their knowledge ofadvanced numerical methods with a focus on geoscience applications, as well as practitioners of the field.

• Polyhedral element methods
• General meshes
• Porous media
• Numerical methods in geomechanics
• Computational techniques for PDEs in complex domains

• Institut Montpelliérain Alexander Grothendieck, University of Montpellier, Montpellier, France

  Daniele Antonio Di Pietro

• MOX-Laboratory for Modeling and Scientific Computing Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

  Luca Formaggia

• Laboratoire de Mathématiques J. A. Dieudonné, Université Côte d’Azur, Nice, France

  Roland Masson

Daniele Di Pietro has been full professor since 2012 and is presently Director of Institut Montpelliérain Alexander Grothendieck at University of Montpellier (France). He is author of 3 research monographs published by Springer and of over 80 scientific papers published in refereed international journals or conference proceedings. He currently serves as associate editor for Numerical Algorithms (Springer). His research fields include the development and analysis of advanced numerical methods for partial differential equations, with applications to fluid and solid mechanics and porous media. Over his career, he has supervised 10 Ph.D. students and 6 post-doctoral fellows.

Luca Formaggia is Full Professor of Numerical Analysis since 2006 at Politecnico di Milano, Italy. He is author of more than 100 publications and Editor of 5 scientific Books published by Springer-Nature and Birkhauser.  His scientific work addresses the study of numerical methods for partial differential equations, scientific computing, computational fluid dynamics with applications to computational geosciences, biomedicine and industrial problems. Currently, he is the President of the Italian Society of Applied and Industrial Mathematics and was the Head of the MOX Laboratory of Politecnico di Milano from 2012 to 2016. Over his career, he has supervised around 40 among PhD students and post-docs. He is the co-Char of the 19th edition of the SIAM Conference on Mathematical and Computational Issues in the Geosciences.

Roland Masson is full professor of Mathematics at the University Côte d’Azur since 2011, member of the Jean-Alexandre Dieudonné department of Mathematics and of the Inria team Coffee. He was previously head of the Applied Mathematics department of the Institut Français du Pétrole (IFP) from 2000 to 2011. He received his PhD in Mathematics from Sorbonne University in 1999 and his Habilitation from Paris East University in 2006. He is an expert in numerical methods and scientific computing for subsurface flow and transport problems. 

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