A posteriori error estimates for variational inequalities: application to a two-phase flow in porous media

Abstract : In this thesis, we consider variational inequalities in the form of partial differential equations with complementarity constraints. We construct a posteriori error estimates for discretizations using the finite element method and the finite volume method, for inexact linearizations employing any semismooth Newton solver and any iterative linear algebraic solver. First, we consider the model problem of contact between two membranes, next we consider its extension into a parabolic variational inequality, and to finish we treat a two-phase compositional flow with phase transition as an industrial application. In the first chapter, we consider the stationnary problem of contact between two membranes. This problem belongs to the wide range of variational inequalities of the first kind. Our discretization is based on the finite element method with polynomials of order p ≥ 1, and we propose two discrete equivalent formulations: the first one as a variational inequality, and the second one as a saddle-point-type problem. We employ the Clarke differential so as to treat the nondifferentiable nonlinearities. It enables us to use semismooth Newton algorithms. Next, any iterative linear algebraic solver is used for the linear system stemming from the discretization. Employing the methodology of equilibrated flux reconstructions in the space H(div,Ω), we get an upper bound on the total error in the energy norm H01(Ω). This bound is fully computable at each semismooth Newton step and at each linear algebraic step. Our estimation distinguishes in particular the three components of the error, namely the discretization error (finite elements), the linearization error (semismooth Newton method), and the algebraic error (GMRES algorithm). We then formulate adaptive stopping criteria for our solvers to ultimately reduce the number of iterations. We also prove, in the inexact semismooth context, the local efficiency property of our estimators, up to a contact term that appears negligeable in numerics. Our numerical experiments illustrate the accuracy of our estimates and the reduction of the number of necessary iterations. They also show the performance of our adaptive inexacte semismooth Newton method. In the second chapter, we are interested in deriving a posteriori error estimates for a parabolic variational inequality and we consider the extension of the model of the first chapter to the unsteady case. We discretize our model using the finite element method of order p ≥ 1 in space and the backward Euler scheme in time. To treat the nonlinearities, we use again semismooth Newton algorithms, and we also employ an iterative algebraic solver for the linear system stemming from the discretization. Using the methodology of equilibrated flux reconstructions in the space H(div,Ω), we obtain, when p=1 and at convergence of the semismooth solver and the algebraic solver, an upper bound for the total error in the energy norm L²(0,T; H01(Ω)). Furthermore, we estimate in this case the time derivative error in a norm close to the energy norm L^2(0,T;H^{-1}(Ω)). In the case p ≥ 1, we present an a posteriori error estimate valid at each semismooth Newton step and at each linear algebraic step in the norm L²(0,T;H01(Ω)). We distinguish in this case the components of the total error, namely the discretization error, the linearization error, and the algebraic error. In particular, it enables us to devise adaptive stopping criteria for our solvers which reduces the number of iterations. In the third chapter, we consider a two-phase liquid-gas compositional (water-hydrogen) flow with hydrogen mass exchange between the phases in porous media. It is a nonlinear system of partial differential equations with nonlinear complementarity constraints which can be interpreted as an evolutive in-time nonlinear variational inequality. We employ the cell-centered finite volume method for the space discretization and the backward Euler scheme for the time discretization. The discretization generates at each time step a nondifferentiable and nonlinear system. As for the previous chapters, we approximate the solution of the nonlinear system stemming from the discretization by an inexact semismooth Newton algorithm. The equilibrated flux reconstructions are obtained in the space H(div,Ω) using the numerical fluxes at the interfaces stemming from the finite volume method, and we reconstruct the phase pressures and the molar fraction in the space H^1(Ω) in order to obtain fully computable upper bound at each time step, semismooth Newton step, and algebraic step. The error measure between the exact solution and the approximate solution is made of the dual norm of the residual supplemented by a residual defined of the complementarity constraints and nonconforming space terms. We finally obtain a posteriori error estimate distinguishing the different error components, namely the discretization error, the linearization error, and the algebraic error. Thus, we formulate adaptive stopping criteria for our solvers in order to reduce the number of iterations. The numerical experiments confirms the benefits of our adaptive inexact semismooth approach.
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Jad Dabaghi. A posteriori error estimates for variational inequalities: application to a two-phase flow in porous media. Numerical Analysis [math.NA]. Sorbonne Université, Université Pierre et Marie Curie, Paris 6, 2019. English. ⟨tel-02151951⟩

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