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1.
P. Chatzipantelidis R.D. Lazarov V. Thomée 《Numerical Methods for Partial Differential Equations》2009,25(3):507-525
We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work 2 . Optimal order convergence may be restored by mesh refinement near the corners of the domain. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
2.
Zhiyue Zhang 《Numerical Methods for Partial Differential Equations》2009,25(2):259-274
In this article, we study the finite volume element methods for numerical solution of the pollution in groundwater flow in a two‐dimensional convex polygonal domain. These type flow are uniform transport in a fully saturated incompressible porous media, which may be anisotropic with respect to hydraulic conductivity, but features a direction independent of dispersivity. A fully finite volume scheme is analyzed in this article. The discretization is defined via a planar mesh consisting of piecewise triangles. Optimal order error estimates in H1 and L2 norms are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
3.
Zhe Yin Hongxing Rui Qiang Xu 《Numerical Methods for Partial Differential Equations》2013,29(3):897-915
A nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium. A sequential implicit time‐stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a modified symmetric finite volume element method and the method of characteristics. Optimal order convergence in H1 and in L2 are proved for full discrete schemes. Finally, some numerical experiments are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
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5.
P. Chatzipantelidis R. D. Lazarov V. Thome 《Numerical Methods for Partial Differential Equations》2004,20(5):650-674
We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L2 and H 1 . The convergence rate in the L∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 相似文献
6.
Upper bounds on the number of determining modes, nodes, and volume elements for a 3D magenetohydrodynamic-$\alpha$ model 下载免费PDF全文
Cung The Anh Nguyen Thi Minh Toai Vu Manh Toi 《Journal of Applied Analysis & Computation》2020,10(2):624-648
In this paper we give upper bounds on the number of determining Fourier modes, determining nodes, and determining volume elements for a 3D MHD-$\alpha$ model. Here the bounds are estimated explicitly in terms of flow parameters, such as viscosity, magnetic diffusivity, smoothing length, forcing and domain size. 相似文献
7.
Min Yang 《Numerical Methods for Partial Differential Equations》2011,27(2):277-291
In this article, we study the a posteriori H1 and L2 error estimates for Crouzeix‐Raviart nonconforming finite volume element discretization of general second‐order elliptic problems in ?2. The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011 相似文献
8.
We give a sufficient and necessary condition that a domain is biholomorphic to the classical domain. 相似文献
9.
Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem
Guoliang He Yinnian He Xinlong Feng 《Numerical Methods for Partial Differential Equations》2007,23(5):1167-1191
A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P1 ? P0 element, the H1 error estimate of optimal order for finite volume solution (uh,ph) is analyzed. And, a uniform H1 error estimate of optimal order for finite volume solution (uh, ph) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
10.
In this paper, first we discuss a technique to compare finite volume method and some well-known finite element methods, namely the dual mixed methods and nonconforming primal methods, for elliptic equations. These both equivalences are exploited to give us a posteriori error estimator for finite volume methods. This estimator is explicitly given, easy to compute and asymptotically exact without any regularity of the solution in unstructured grids. 相似文献
11.
Grigory Panasenko Marie‐Claude Viallon 《Mathematical Methods in the Applied Sciences》2013,36(14):1892-1917
The method of asymptotic partial domain decomposition has been proposed for partial differential equations set in rod structures, depending on a small parameter. It reduces the dimension of the problem (or simplifies it in another way) in the main part of the domain keeping the initial formulation in the remaining part and prescribing the asymptotically precise conditions on the interface. This paper is devoted to the finite volume implementation of the method of asymptotic partial domain decomposition. We consider a model problem in a thin domain (its thickness is a small parameter). We obtain an error estimate, expressed in terms of the small parameter and the step of the mesh. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
12.
Extension theorems for plate elements are established. Their applications to the analysis of nonoverlapping domain decomposition methods for solving the plate bending problems are presented. Numerical results support our theory.
13.
A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes 下载免费PDF全文
Zhiqiang Sheng Jingyan Yue Guangwei Yuan 《Numerical Methods for Partial Differential Equations》2017,33(6):2159-2178
Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017 相似文献
14.
Fluid flow in naturally fractured porous media can always be regarded as an unbounded domain problem and be better solved by finite/infinite elements. In this paper, a three-dimensional two-direction mapped infinite element is generated and combined with conventional finite elements and one direction infinite element to simulate poroelasticity. Therefore, the entire semi-infinite domain can be included in the numerical analysis. Both single- and dual-porosity porous media are considered. For the purpose of validation, we compare the results of finite/infinite elements with those of finite elements under two extreme boundary conditions. The comparison indicated that mapped infinite element is an appropriate approach to model fluid flow in porous media and provides an intermediate solution. 相似文献
15.
《Applied Mathematical Modelling》2014,38(7-8):2265-2279
This paper details the evaluation and enhancement of the vertex-centred finite volume method for the purpose of modelling linear elastic structures undergoing bending. A matrix-free edge-based finite volume procedure is discussed and compared with the traditional isoparametric finite element method via application to a number of test-cases. It is demonstrated that the standard finite volume approach exhibits similar disadvantages to the linear Q4 finite element formulation when modelling bending. An enhanced finite volume approach is proposed to circumvent this and a rigorous error analysis conducted. It is demonstrated that the developed finite volume method is superior to both standard finite volume and Q4 finite element methods, and provides a practical alternative to the analysis of bending-dominated solid mechanics problems. 相似文献
16.
Norikazu Saito 《Numerical Functional Analysis & Optimization》2013,34(4):501-527
The L 2-penalty fictitious domain method is based on a reformulation of the original problem in a larger simple-shaped domain by introducing a discontinuous reaction term with a penalty parameter ε > 0. We first derive regularity results and some a priori estimates and then prove several error estimates. We also give several error estimates for discretization problems by the finite element and finite volume methods. 相似文献
17.
T.V. Voitovich S. Vandewalle 《Numerical Methods for Partial Differential Equations》2007,23(5):1059-1082
This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high‐order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
18.
多孔介质中可压缩可混溶驱动问题的有限体积元法 总被引:2,自引:0,他引:2
马克颖 《高校应用数学学报(A辑)》2005,20(2):161-169
有界区域上多孔介质中可压缩可混溶驱动问题由两个非线性抛物型方程耦合而成:压力方程和饱和度方程均是抛物型方程.运用有限体积元法对两个方程进行数值分析,给出了全离散有限体积元格式,并通过详细的理论分析,得到了近似解与原问题真解的最优H^1模误差估计。 相似文献
19.
Venera
Khoromskaia Boris N. Khoromskij Felix Otto 《Numerical Linear Algebra with Applications》2020,27(3)
We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned conjugate gradient iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver, we numerically investigate the convergence of the representative volume element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in Gloria et al. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE. 相似文献
20.
《Numerical Methods for Partial Differential Equations》2018,34(2):661-685
Numerical simulation of oil‐water two‐phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection‐diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order in norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems. 相似文献