Finite volume element approximation of the coupled continuum pipe‐flow/Darcy model for flows in karst aquifers

A finite volume element method is applied to approximate the continuum pipe‐flow/Darcy problem, which models the coupled conduit flow and porous media flow in Karst aquifers. A decoupled scheme is proposed for solving the coupled discretization problem. Optimal error estimates in L² norm and H¹ norm are given in this article. Some numerical examples are presented to verify the theoretical results and demonstrate the effectiveness of the decoupling approach. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 376–392, 2014     

Error estimates of finite volume element method for the pollution in groundwater flow

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     

A high order finite volume method for one dimensional nonlocal reactive flows of parabolic type

In this paper, we consider a high order finite volume approximation of one‐dimensional nonlocal reactive flows of parabolic type. The method is obtained by discretizing in space by arbitrary order vertex‐centered finite volumes, followed by a modified Simpson quadrature scheme for the time stepping. Compared to the existed finite volume methods, this new finite volume scheme could achieve the desired accuracy with less data storage by employing higher‐order trial spaces. The finite volume approximations are proved to possess optimal order convergence rates in the H¹‐norm and L²‐norm, which are also confirmed by numerical tests. Copyright © 2015 John Wiley & Sons, Ltd.     

A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure‐velocity equation and the concentration equation. In this article, we present a mixed finite volume element method for the approximation of pressure‐velocity equation and a discontinuous Galerkin finite volume element method for the concentration equation. A priori error estimates in L^∞(L²) are derived for velocity, pressure, and concentration. Numerical results are presented to substantiate the validity of the theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Finite volume element method for monotone nonlinear elliptic problems

In this article, we consider the finite volume element method for the monotone nonlinear second‐order elliptic boundary value problems. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz‐continuous, and with the minimal regularity assumption on the exact solution, that is, u∈H¹(Ω), we show that the finite volume element method has a unique solution, and the finite volume element approximation is uniformly convergent with respect to the H¹ ‐norm. If u∈H^1+ε(Ω),0 < ε ≤ 1, we develop the optimal convergence rate \begin{align*}\mathcal{O}(h^{\epsilon})\end{align*} in the H¹ ‐norm. Moreover, we propose a natural and computationally easy residual‐based H¹ ‐norm a posteriori error estimator and establish the global upper bound and local lower bounds on the error. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error estimates for finite volume element methods for general second‐order elliptic problems

We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H¹ error estimate, H¹ superconvergence and L^p (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003.     

Numerical analysis of finite element methods for miscible displacements in porous media

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element ( SUPG ) method is applied to the convection–diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 519–548, 1998     

多孔介质中可压缩可混溶驱动问题的有限体积元法

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。     

Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems

In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approximating the solution of a class of second‐order linear elliptic problems is discussed. Optimal error estimates in L² and broken H¹‐ norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both and ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016     

Lp error estimates and superconvergence for covolume or finite volume element methods

We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the L^p norm, 2 ≤ p ≤ ∞, are derived. We also show second‐order convergence in the L^p norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the “supercloseness” results in Chou and Li [Math Comp 69(229) (2000), 103–120] to the L^p based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463–486, 2003     

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L^∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

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     

Analysis of an Euler implicit‐mixed finite element scheme for reactive solute transport in porous media

In this article, we analyze an Euler implicit‐mixed finite element scheme for a porous media solute transport model. The transporting flux is not assumed given, but obtained by solving numerically the Richards equation, a model for subsurface fluid flow. We prove the convergence of the scheme by estimating the error in terms of the discretization parameters. In doing so we take into account the numerical error occurring in the approximation of the fluid flow. The article, is concluded by numerical experiments, which are in good agreement with the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Parabolic finite volume element equations in nonconvex polygonal domains

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     

An approximation of incompressible miscible displacement in porous media by mixed finite elements and symmetric finite volume element method of characteristics

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     

Finite element analysis of thermally coupled nonlinear Darcy flows

We consider a coupled system describing nonlinear Darcy flows with temperature dependent viscosity and with viscous heating. We first establish existence, uniqueness, and regularity of the weak solution of the system of equations. Next, we decouple the coupled system by a fixed point algorithm and propose its finite element approximation. Finally, we present convergence analysis with an error estimate between continuous solution and its iterative finite element approximation.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Error estimates for mixed finite element approximations of the viscoelasticity wave equation

This paper studies mixed finite element approximations to the solution of the viscoelasticity wave equation. Two new transformations are introduced and a corresponding system of first‐order differential‐integral equations is derived. The semi‐discrete and full‐discrete mixed finite element methods are then proposed for the problem based on the Raviart–Thomas–Nedelec spaces. The optimal error estimates in L²‐norm are obtained for the semi‐discrete and full‐discrete mixed approximations of the general viscoelasticity wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element approximation of coupled seismic and electromagnetic waves in fluid‐saturated poroviscoelastic media

This work presents a collection of global and iterative finite element procedures for the numerical approximation of coupled seismic and electromagnetic waves in 2D bounded fluid‐saturated porous media, with absorbing boundary conditions at the artificial boundaries. The equations being analyzed are the coupled Biot's equations of motion and Maxwell equations in the diffusive range. Both seismoelectric and electroseismic coupling are simultaneously included and analyzed in the model. The case of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM‐mode) is analyzed in detail, including the derivation of a priori error estimates on the global finite element procedure and results on the convergence of a domain decomposition iterative algorithm. Later, the corresponding results for the case of horizontally polarized shear waves coupled with the transverse electric polarization (SHTE‐mode) are stated. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A discontinuous finite volume element method for second‐order elliptic problems

In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L² ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012