Gradient Discretization Method Research Articles

Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to [F. Bonaldi, K. Brenner, J. Droniou and R. Masson, Comput. Math. with Appl. 98 (2021)], the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in [K. Brenner, J. Hennicker, R. Masson and P. Samier, J. Comput. Phys. 357 (2018)], [J. Aghili, K. Brenner, J. Hennicker, R. Masson and L. Trenty, GEM – Int. J. Geomath. 10, (2019)], unlike single-phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface. The model is discretized using the gradient discretization method [J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, Springer, Mathematics & Applications, 82 (2018)], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of [F. Bonaldi, K. Brenner, J. Droniou and R. Masson, Comput. Math. with Appl. 98 (2021)] is extended to the discontinuous pressure model and the convergence to a weak solution is proved. Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and ℙ2 finite elements for the mechanics.

Gradient discretization of two-phase flows coupled with mechanical deformation in fractured porous media

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is also coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. The model is discretized using the gradient discretization method [30], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. Here, we describe the model together with its numerical discretization and, using discrete compactness techniques, we prove a convergence result (up to a subsequence) assuming the non-degeneracy of the phase mobilities and that the discrete solutions remain physical in the sense that, roughly speaking, the porosity does not vanish and the fractures remain open. This is, to our knowledge, the first convergence result for this type of model taking into account two-phase flows in fractured porous media and the non-linear poromechanical coupling. Previous related works consider a linear approximation obtained for a single phase flow by freezing the fracture conductivity [41,42]. Numerical tests employing the Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and P2 finite elements for the mechanical deformation are also provided to illustrate the behavior of the solution to the model.

Design and convergence analysis of numerical methods for stochastic evolution equations with Leray–Lions operator

Abstract The gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray–Lions type operator. The problem contains the stochastic $p$-Laplace equation as a particular case. The convergence of the gradient scheme (GS) solutions is proved by using discrete functional analysis techniques, Skorohod theorem and the Kolmogorov test. In particular, we provide an independent proof of the existence of weak martingale solutions for the problem. In this way we lay foundations and provide techniques for proving convergence of the GS approximating stochastic partial differential equations.

Note on a [formula omitted]-error estimate of a nonlinear finite volume scheme for a semi-linear heat equation

We consider, as discretization in space, the nonconforming mesh developed in SUSHI (Scheme Using Stabilization and Hybrid Interfaces) developed in Eymard et al. (2010) for a semi-linear heat equation. The time discretization is performed using a uniform mesh. We are concerned with a nonlinear scheme that has been studied in Bradji (2016) in the context of the general framework GDM (Gradient Discretization Method) (Droniou et al., 2018) which includes SUSHI. We provide sufficient conditions on the size of the spatial mesh and the time step which allow to prove a W1,∞(L2)-error estimate. This error estimate can be viewed as an improvement for the W1,2(L2)-error estimate proved in Bradji (2016). The W1,∞(L2)-error estimate we want to prove in this note was stated without proof in Bradji (2016, Remark 7.2, Page 1302). Its proof is based on a comparison with an appropriately chosen auxiliary finite volume scheme along with the derivation of some new estimates on its solution.

The Gradient Discretization Method for Slow and Fast Diffusion Porous Media Equations

The gradient discretization method (GDM) is a generic framework for designing and analyzing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion models, and a concentration-dependent diffusion tensor. Using discrete functional analysis techniques, we establish a strong $L^2$-convergence of the approximate gradients and a uniform-in-time convergence for the approximate solution, without assuming nonphysical regularity assumptions on the data or continuous solution. Being established in the generic GDM framework, these results apply to a variety of numerical methods, such as finite volume, (mass-lumped) finite elements, etc. The theoretical results are illustrated, in both fast and slow diffusion regimes, by numerical tests based on two methods that fit the GDM framework: mass-lumped conforming $\mathbb{P}_1$ finite elements and the hybrid mimetic mixed method.

A new analysis for the convergence of the gradient discretization method for multidimensional time fractional diffusion and diffusion-wave equations

We consider as discretization in space, the GDM (Gradient Discretization Method) developed recently in Droniou et al. (2018), to approximate multidimensional time fractional diffusion and diffusion-wave equations where the fractional order α of the time derivative is respectively satisfying 0<α<1 and 1<α<2. The time fractional derivative is given in the Caputo sense. The time discretization is performed using a uniform mesh. For the time fractional diffusion equation, we derive an implicit scheme. In addition to an L∞(L2)-a priori estimate, which can be derived from a reasoning in Bradji (2018), we present and prove a new L2(H01)-a priori estimate for the discrete problem. Under the assumption that the exact solution is sufficiently smooth, these a priori estimates allow to prove error estimates in discrete norms of L2(H01) and L∞(L2). The convergence order in these estimates is optimal in the sense of two points of view. The first one is that the order in space is the same one proved in Droniou et al. (2018) for elliptic equation, whereas the second point of view is that the particular case of this order when α=1 is the same one obtained in Droniou et al. (2018) for the standard heat equation. For the time fractional diffusion-wave equation, we derive two implicit schemes. A full convergence analysis is carried out for both schemes. In particular, we develop new a priori estimates which yield error estimates in several discrete norms for each scheme. The convergence is proved to be unconditional and optimal. The convergence results are obtained thanks to a comparison with appropriately chosen auxiliary gradient schemes and to the stated a priori estimates. These results improve the ones of Bradji (2017) in which a conditional convergence is proved for a SUSHI (Scheme using Stabilization and Hybrid Interfaces) approximating a time-fractional diffusion-wave equation. For both cases of time fractional diffusion and diffusion-wave equations, we show the well-posedness in the sense that the discrete solutions depend continuously on the data of the considered problems. The stated results can be extended to multi-term time-fractional diffusion and diffusion-wave equations. One of the main features when using GDM is that their results hold for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes. Some examples of schemes recovered by the GSs (Gradient Schemes) presented in this work are sketched. Among these examples, we quote the one presented in Jin et al. (2015) to approximate time fractional diffusion equations. This work extends and improves some results presented in brief or stated without proof in the notes Bradji (2018). We present some numerical tests using SUSHI introduced in Eymard et al. (2010) to support the theoretical results.

A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations

A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gradient regions. Using the gradient discretization framework we prove convergence of the scheme for linear and quasilinear equations under minimal regularity assumptions. The error due to artificial boundary conditions is also analyzed, shown to be of higher order and shown to depend only locally on the regularity of the solution. Numerical experiments illustrate our theoretical findings and the local method’s accuracy is compared against the non local approach.

Notes on the convergence order of gradient schemes for time fractional differential equations

We apply the GDM (Gradient Discretization Method), developed recently, as discretization in space to time-fractional diffusion and diffusion-wave equations with a fractional derivative of Caputo type in any space dimension.In the case of time-fractional diffusion equations, we establish an implicit scheme, and we prove an L∞(L2)-error estimate. A similar result in a discrete L∞(H01)–norm is also stated.To construct the numerical scheme for the time-fractional diffusion-wave equation, we write the equation in the form of a system of two low-order equations. We state an a prior estimate result that helps us to derive error estimates in discrete semi-norms of L∞(H1) and H1(L2). The convergence is unconditional. Another gradient scheme is also suggested. We state its convergence results, which improve the convergence order proved recently for a SUSHI scheme.These results hold then for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes.

A Gradient Discretization Method to Analyze Numerical Schemes for Nonlinear Variational Inequalities, Application to the Seepage Problem

Using the gradient discretisation method (GDM), we provide a complete and unified numerical analysis for non-linear variational inequalities (VIs) based on Leray--Lions operators and subject to non-homogeneous Dirichlet and Signorini boundary conditions. This analysis is proved to be easily extended to the obstacle and Bulkley models, which can be formulated as non-linear VIs. It also enables us to establish convergence results for many conforming and nonconforming numerical schemes included in the GDM, and not previously studied for these models. Our theoretical results are applied to the hybrid mimetic mixed method (HMM), a family of schemes that fit into the GDM. Numerical results are provided for HMM on the seepage model, and demonstrate that, even on distorted meshes, this method provides accurate results.

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.