An adaptive discontinuous finite volume method for elliptic problems

Liu, Jiangguo; Mu, Lin; Ye, Xiu

doi:10.1016/j.cam.2011.05.051

Cited by 21 publications

(13 citation statements)

References 14 publications

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“…The finite volume method possesses a special feature of the local conservativity of the numerical fluxes, and is becoming more and more popular. See, for instance, [27] for degenerate parabolic problems, [20] for hyperbolic problems, [21] for elliptic problems, and [37,9] for HJB equations.…”

Section: Shuhua Zhang Xinyu Wang and Hua LImentioning

confidence: 99%

Modeling and computation of water management by real options

Zhang¹,

Chang²,

Hua³

2018

Journal of Industrial &Amp; Management Optimization

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It becomes increasingly important to manage water and improve the efficiency of irrigation under higher temperatures and irregular precipitation patterns. The choice of investment in water saving technologies and its timing play key roles in improving efficiency of water use. In this paper, we use a real option approach to establish a model to handle future uncertainties about the water price. In addition, to match the practical situation, the expiration of the real option is considered to be finite in our model, such that it is difficult to solve the model. Therefore, we reformulate the problem into a linear parabolic variational inequality (VI) and develop a power penalty method to solve it numerically. Thus, a nonlinear partial differential equation (PDE) is obtained, which is shown to be uniquely solvable and the solution of the nonlinear PDE converges to that of the VI at the rate of O(λ − k 2) with λ being the penalty number. Furthermore, a so-called fitted finite volume method is proposed to solve the nonlinear PDE. Finally, several numerical experiments are performed. It is shown that the subjective discount rate will affect the investment boundary mostly, and the flexibility to suspend operation will enlarge the investment region.

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Section: Shuhua Zhang Xinyu Wang and Hua LImentioning

confidence: 99%

Modeling and computation of water management by real options

Zhang¹,

Chang²,

Hua³

2018

Journal of Industrial &Amp; Management Optimization

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“…In their research, FVM was used to discretize the governing equations while FEM was applied to determine the gradient quantities at cell faces. The adaptive discontinuous FVM was presented by Liu et al (2011) to solve the elliptic boundary value problems on triangular meshes. They utilized residual a posteriori error estimation in the adaptive refinement procedure.…”

Section: Introductionmentioning

confidence: 99%

A Proposed Damping Coefficient of Quick Adaptive Galerkin Finite Volume Solver for Elasticity Problems

Yazdi¹,

Amiri²,

Gharebaghi³

2019

JSSCM

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A Quick Adaptive Galerkin Finite Volume (QAGFV) solution of Cauchy momentum equations for plane elastic problems is presented in this research. A new damping coefficient is introduced to preserve the efficiency of the iterative pseudo-explicit solution procedure. It is shown that the numerical oscillations are not only effectively damped by the proposed damping coefficient, but also that the rate of the convergence of QAGFV algorithm increases. Furthermore, the numerical results show that the proposed coefficient is not sensitive to the spatial discretization. In order to improve the accuracy of the computed stress and displacement fields, an automatic twodimensional h-adaptive mesh refinement procedure is adopted for shape-function-free solution of the governing equations. For verification, two classical problems and their analytical solutions have been investigated. The first is a uniaxial loaded plate with holes, and the second is a cantilever beam under a concentrated load. The results show a good agreement between QAGFV and analytical method. Moreover, the direct and iterative approaches of the finite element method have been implemented in FORTRAN to evaluate the efficiency and accuracy of the presented algorithm. In the end, the corresponding results of some problems have been compared to the QAGFV solutions. The results confirm that the presented h-adaptive QAGFV solver is accurate and highly efficient especially in a large computational domain.

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“…Finite volume method, as an important numerical tool for solving partial differential equations, has been widely used in the engineering community for fluid computations (see [9,14,15,29,30]). Finite volume method is intuitive since it is based on local conservation of mass, momentum, and energy over volumes.…”

Section: Introductionmentioning

confidence: 99%

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

Zhang

2018

Bound Value Probl

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In this paper, based on the pressure project method, we consider an adaptive stabilized finite volume method for the Oseen equations with the lowest equal order finite element pair. Firstly, we develop the discrete forms in both finite element and finite volume methods, and establish the existence and uniqueness of numerical solutions by establishing the equivalence of linear terms in finite element and finite volume methods. Secondly, a residual type a posteriori error estimator is designed, and the computable global upper and local lower bounds between the exact solutions and the finite volume solutions are established. Thirdly, a discrete local lower bound between two successive finite volume solutions is obtained, convergence analysis of the adaptive stabilized finite volume method is also performed. Finally, some numerical results are presented to verify the performances of the developed error estimators and confirm the established theoretical findings. MSC: 65N15; 65N30; 76D05

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An adaptive discontinuous finite volume method for elliptic problems

Cited by 21 publications

References 14 publications

Modeling and computation of water management by real options

Modeling and computation of water management by real options

A Proposed Damping Coefficient of Quick Adaptive Galerkin Finite Volume Solver for Elasticity Problems

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

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