Ph.D. thesis

presented to obtain the degree of Doctor of Sciences of the University Paris XI, specialization Mathematics
&
Doctor of the Czech Technical University in Prague, specialization Mathematical Modeling by

Martin Vohralík

Title
NUMERICAL METHODS FOR NONLINEAR ELLIPTIC AND PARABOLIC EQUATIONS
Application to flow problems in porous and fractured media

Chapters

  1. Combined finite volume–finite element schemes for degenerate parabolic convection–reaction–diffusion problems
  2. Discrete Poincaré–Friedrichs inequalities
  3. Equivalence between lowest-order mixed finite element and multi-point finite volume methods
  4. Mixed and nonconforming finite element methods on a fracture network

Full text
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Slides
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Supervisors
Danielle HILHORST (CNRS & Université Paris XI, France)   and   Jiří MARYŠKA (Technical University of Liberec, Czech Republic)

Defended
Thursday, December 9 2004 at 4 p.m., room 113–115 of building 425, University Paris XI, Orsay, France

Jury Referees Key words

AMS subject classifications
65M12, 65N30, 76M10, 76M12, 76S05, 35J20, 35K65, 46E35

Abstract

This thesis deals with numerical methods for the discretization of nonlinear elliptic and parabolic convection–reaction–diffusion partial differential equations. We analyze these methods and apply them to the effective simulation of flow and contaminant transport in porous and fractured media.

In Chapter 1 we propose a scheme allowing for efficient, robust, conservative, and stable discretizations of nonlinear degenerate parabolic convection–reaction–diffusion equations on unstructured grids in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, by means of the nonconforming or mixed-hybrid finite element method and the other terms by means of the finite volume method. The essential part of this chapter is then devoted to showing the existence and uniqueness of a discrete solution and its convergence to a weak solution of the continuous problem. The proofs permit in particular to avoid restrictive hypotheses on the mesh often used in the literature. We finally propose a version of this scheme for nonmatching grids, combining this time the finite volume method with the piecewise linear conforming finite element method. We then apply this version to contaminant transport simulations in porous media.

In Chapter 2 we present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H1, indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. The results are important in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.

In Chapter 3 we show that the lowest-order Raviart–Thomas mixed finite element method for elliptic problems on simplicial meshes in two or three space dimensions is equivalent to a particular multi-point finite volume scheme. We study this scheme and apply it to the discretization of nonlinear parabolic convection–reaction–diffusion equations. This approach allows significant reduction of the computational time of the mixed finite element method without any loss of its high precision, which is confirmed by numerical experiments.

Finally, in Chapter 4 we propose a version of the lowest-order Raviart–Thomas mixed finite element method for the approximation of elliptic problems on a system of two-dimensional polygons placed in three-dimensional space, prove that it is well-posed, and study its relation to the nonconforming finite element method. These results are finally applied to the simulation of underground water flow through a system of polygons representing a network of fractures that perturbs a rock massif.