检索结果-内蒙古大学图书馆

A Conservative gradient discretization method for Parabolic Equations

Advances in Applied Mathematics and Mechanics 2021年第1期13卷 232-260页

作者： Huifang Zhou Zhiqiang Sheng Guangwei Yuan The Graduate School of China Academy of Engineering Physics BeijingChina Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijingChina

In this paper,we propose a new conservative gradient discretization method(GDM)for one-dimensional parabolic partial differential equations(PDEs).We use the implicit Euler method for the temporal discretization and conservative gradient discretization method for spatial *** method is based on a new cellcentered meshes,and it is locally *** has smaller truncation error than the classical finite volume method on uniform *** use the framework of the gradient discretization method to analyze the stability and *** numerical experiments show that the new method has second-order ***,it is more accurate than the classical finite volume method in flux error,L2 error and L¥error.

关键词： gradient discretization method mass conservation parabolic equations

来源：评论

学校读者我要写书评

暂无评论

THE gradient discretization method FOR SLOW AND FAST DIFFUSION POROUS MEDIA EQUATIONS

引用

SIAM JOURNAL ON NUMERICAL ANALYSIS 2020年第3期58卷 1965-1992页

作者： Droniou, Jerome Le, Kim-Ngan Monash Univ Sch Math Clayton Vic 3800 Australia

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 P-1 finite elements and the hybrid mimetic mixed method.

关键词： gradient discretization method fast-slow diffusion models porous media equations numerical methods convergence analysis

来源：评论

学校读者我要写书评

暂无评论

THE gradient discretization method FOR OPTIMAL CONTROL PROBLEMS, WITH SUPERCONVERGENCE FOR NONCONFORMING FINITE ELEMENTS AND MIXED-HYBRID MIMETIC FINITE DIFFERENCES

引用

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第6期55卷 3640-3672页

作者： Droniou, Jerome Nataraj, Neela Shylaja, Devika Monash Univ Sch Math Sci Clayton Vic 3800 Australia Indian Inst Technol Dept Math Mumbai 400076 Maharashtra India Indian Inst Technol IITB Monash Res Acad Mumbai 400076 Maharashtra India

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretization 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 are discussed. These superconvergence results are shown to apply to nonconforming 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.

关键词： elliptic equations optimal control numerical schemes error estimates superconvergence gradient discretization method gradient schemes nonconforming P-1 finite elements mimetic finite differences

来源：评论

学校读者我要写书评

暂无评论

NUMERICAL ANALYSIS OF A MIXED-DIMENSIONAL POROMECHANICAL MODEL WITH FRICTIONLESS CONTACT AT MATRIX-FRACTURE INTERFACES

引用

MATHEMATICS OF COMPUTATION 2024年第349期93卷 2103-2134页

作者： Bonaldi, Francesco Droniou, Jerome Masson, Roland Univ Montpellier IMAG CNRS Montpellier France Univ Perpignan Via Domitia LAMPS Perpignan France Monash Univ Sch Math Clayton Vic 3800 Australia Univ Cote Dazur Team Coffee Inria CNRSLab JA Dieudonne Nice France

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid -dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the gradient discretization framework (see J. Droniou, R. Eymard, T. Galloue center dot t, C. Guichard, and R. Herbin [The gradient discretisation method, Springer, Cham, 2018]), encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second -order finite elements (P2) for the mechanical displacement coupled with face -wise constant (P0) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

关键词： Poromechanics discrete fracture matrix models contact mechanics Darcy flow discontinuous pressure model gradient discretization method convergence analysis

来源：评论

学校读者我要写书评

暂无评论

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

引用

IMA JOURNAL OF NUMERICAL ANALYSIS 2022年第2期42卷 1143-1179页

作者： Droniou, Jerome Goldys, Beniamin Le, Kim-Ngan Monash Univ Sch Math Clayton Vic 3800 Australia Univ Sydney Sch Math & Stat Sydney NSW 2006 Australia Univ Sydney Univ Sydney Nano Inst Sydney NSW 2006 Australia

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.

关键词： p-Laplace equation stochastic PDE numerical methods gradient discretization method convergence analysis

来源：评论

学校读者我要写书评

暂无评论

Convergence analysis for a nonlinear system of parabolic variational inequalities

引用

JOURNAL OF INEQUALITIES AND APPLICATIONS 2022年第1期2022卷 1-18页

作者： Alnashri, Yahya Umm Al Qura Univ Al Qunfudah Univ Coll Dept Math Mecca Saudi Arabia

This work aims to provide a comprehensive and unified numerical analysis for a nonlinear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish the existence of an exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and nonconforming numerical schemes.

关键词： Nonlinear parabolic variational inequalities PDEs gradient discretization method gradient schemes Obstacle problem Convergence analysis Nonconforming P1 finite element method

来源：评论

学校读者我要写书评

暂无评论

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

引用

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS 2021年第5期55卷 1741-1777页

作者： Bonaldi, Francesco Brenner, Konstantin Droniou, Jerome Masson, Roland Pasteau, Antoine Trenty, Laurent Univ Cote Azur INRIA CNRS Lab JA DieudonneTeam Coffee Nice France Monash Univ Sch Math Clayton Vic 3800 Australia Andra Chatenay Malabry France

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. Gallouet, 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 sc

关键词： Poro-mechanics discrete fracture matrix models two-phase Darcy flows discontinuous pressure model gradient discretization method convergence analysis

来源：评论

学校读者我要写书评

暂无评论

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

引用

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2021年 98卷 40-68页

作者： Bonaldi, Francesco Brenner, Konstantin Droniou, Jerome Masson, Roland Univ Cote dAzur Lab JA Dieudonne Team Coffee CNRS Nice France Monash Univ Sch Math Clayton Vic 3800 Australia

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.

关键词： Poromechanics Discrete fracture matrix models Two-phase Darcy flows gradient discretization method Convergence analysis

来源：评论

学校读者我要写书评

暂无评论

HIGH-ORDER MASS-LUMPED SCHEMES FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

引用

SIAM JOURNAL ON NUMERICAL ANALYSIS 2020年第1期58卷 153-188页

作者： Droniou, Jerome Eymard, Robert Monash Univ Sch Math Melbourne Vic Australia Univ Paris Est Marne la Vallee Lab Anal & Math Appl Champs Sur Marne France

We present and analyze a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regularity assumption on the solution. A detailed analysis is then performed to understand the design properties that enable a scheme, despite these piecewise constant approximations and the degeneracy of the model, to satisfy high-order error estimates if the solution is piecewise smooth. Numerical tests, based on continuous and discontinuous approximation methods, are provided on a variety of one- and two-dimensional problems, showing the influence on the convergence rate of the nature of the degeneracy and of the design choices.

关键词： nonlinear degenerate elliptic equations gradient discretization method error estimate mass-lumping finite elements discontinuous Galerkin Stefan problem porous medium equation numerical scheme

来源：评论

学校读者我要写书评

暂无评论

Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model 8th

Uniform-in-Time Convergence of Numerical Schemes for a Two-P...

引用

8th International Symposium on Finite Volumes for Complex Applications (FVCA)

作者： Droniou, J. Hennicker, J. Masson, R. Monash Univ Sch Math Sci Clayton Vic 3800 Australia Univ Cote dAzur INRIA CNRS Lab JA DieudonneTeam Coffee Nice France TOTAL SA CSTJF Ave Larribau F-64018 Pau France

ISBN: (纸本)9783319573977;9783319573960

Flowand transport in fractured porous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

关键词： Discrete fracture model Two-phase Darcy flow Uniform-in-time convergence gradient discretization method

来源：评论

学校读者我要写书评

暂无评论

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案：

请选择收藏分类：

通借通还

建议与咨询 留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

时间限定

文献类型

馆藏选择

核心期刊

语言

文献类型

帮助

文字说明：

检索规则说明：

检索范例：

分类表

所选分类

限定检索结果

文献类型

馆藏范围

日期分布

学科分类号

主题

机构

作者

语言

请选择保存的检索档案： 新增检索档案 确定 取消

请选择收藏分类： 新增自定义分类 确定 取消

通借通还

建议与咨询留下您的常用邮箱和电话号码，以便我们向您反馈解决方案和替代方法

请选择保存的检索档案：

请选择收藏分类：