Finite Volume Element Formulation Research Articles

Finite Volume Element Approximation for the Elliptic Equation with Distributed Control

In this paper, we consider a priori error estimates for the finite volume element schemes of optimal control problems, which are governed by linear elliptic partial differential equation. The variational discretization approach is used to deal with the control. The error estimation shows that the combination of variational discretization and finite volume element formulation allows optimal convergence. Numerical results are provided to support our theoretical analysis.

A Stabilized Crank–Nicolson Mixed Finite Volume Element Formulation for the Non-stationary Incompressible Boussinesq Equations

At first, a semi-discrete Crank---Nicolson (CN) formulation with respect to time for the non-stationary incompressible Boussinesq equations is presented. Then, a fully discrete stabilized CN mixed finite volume element (SCNMFVE) formulation based on two local Gauss integrals and parameter-free is established directly from the semi-discrete CN formulation with respect to time. Next, the error estimates for the fully discrete SCNMFVE solutions are derived by means of the standard CN mixed finite element method. Finally, some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results, the computing load for the fully discrete SCNMFVE formulation are far fewer than that for the stabilized mixed finite volume element (SMFVE) formulation with the first time accuracy, and its accumulation of truncation errors in the computational process is far lesser than that of the SMFVE formulation with the first time accuracy. Thus, the advantage of the fully discrete SCNMFVE formulation for the non-stationary incompressible Boussinesq equations is shown sufficiently.

A reduced-order FVE extrapolation algorithm based on proper orthogonal decomposition technique and its error analysis for Sobolev equation

In this paper, a proper orthogonal decomposition (POD) technique is used to treat the classical finite volume element (FVE) formulation for two-dimensional (2D) Sobolev equation. A reduced-order FVE extrapolation algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD technique is established for 2D Sobolev equation. The error estimates with respect to the norm in $$H_0^1(\Omega )$$ between the reduced-order FVE extrapolation algorithm solutions and the classical FVE solutions are provided for 2D Sobolev equation. The implementation for solving the reduced-order FVE extrapolation algorithm is given. By comparing the numerical results of the reduced-order FVE extrapolation algorithm, the classical FVE formulation, reduced-order finite element (FE) formulation, classical FE formulation, reduced-order finite difference (FD) scheme, and classical FD scheme for 2D Sobolev equation, it is shown that the reduced-order FVE extrapolation algorithm is one of the most effective numerical methods. Moreover, it is shown that the reduced-order FVE extrapolation algorithm based on POD technique is feasible and efficient for solving 2D Sobolev equation.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

A fully discrete stabilized mixed finite volume element formulation for the non-stationary conduction–convection problem

At first, a semi-discrete formulation with respect to time for the non-stationary conduction–convection problem is recalled. Then, a fully discrete stabilized mixed finite volume element (SMFVE) formulation based on two local Gauss integrals and parameter-free is established directly from the semi-discrete formulation with respect to time for the non-stationary conduction–convection problem. Following this, the error estimates for the fully discrete SMFVE solutions are derived by means of the standard mixed finite element method. Finally, some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results, the degrees of freedom of the fully discrete SMFVE formulation are far fewer than those of the classical finite volume element (FVE) formulation without any stabilization, and its numerical solutions are more stable than those of the classical FVE formulation without any stabilization, thus validating that the fully discrete SMFVE formulation is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem.

A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine the classical finite volume element (FVE) method with the POD method to obtain a reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional solute transport problems, which have real life practical applications. We then provide error estimates between the reduced-order POD FVE solutions and classical FVE solutions and we provide implementation of an extrapolation algorithm for solving the reduced-order FVE formulation. Thus, we provide the theoretical basis for practical applications. A numerical example is then used to ascertain that the results of numerical computation are consistent with the theoretical derivations. Moreover, it is shown that the reduced-order FVE formulation based on POD method is both feasible and efficient for solving two-dimensional solute transport problems.

A reduced-order Crank–Nicolson finite volume element formulation based on POD method for parabolic equations

In this paper, a proper orthogonal decomposition (POD) method is used to deal with a classical Crank–Nicolson finite volume element (CNFVE) method for two-dimensional parabolic equations. A reduced-order CNFVE formulation with lower dimensions and sufficiently high accuracy based on POD technique is established, the error estimates between reduced-order CNFVE solutions based on the POD method and classical CNFVE solutions are provided, and the extrapolation algorithm for solving reduced-order CNFVE formulation is implemented. Some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions. Moreover, it is shown that the reduced-order CNFVE formulation based on POD method is feasible and efficient for solving two-dimensional parabolic equations.

非定常Stokes方程的稳定化CN有限体积元格式

建立二维非定常Stokes方程的时间二阶精度的稳定化Crank-Nicolson (CN)有限体积元格式, 并给出其稳定化CN有限体积元解的误差估计。数值实验说明时间二阶精度的稳定化CN有限体积元格式比时间一阶精度格式更优越, 从而表明稳定化CN有限体积元格式对于求解非定常Stokes方程的数值解是有效可行的。 A stabilized Crank-Nicolson (CN) finite volume element formulation with time second-order accuracy is established for two-dimensional non-stationary Stokes equation. The error estimates of its numerical solutions are provided. Some numerical experiments are presented illustrating that the stabilized CN finite volume element formulation with time second-order accuracy is far more advantageous than that with time first-order accuracy, thus validating that the stabilized CN finite volume element formulation is feasible and efficient for finding the numerical solutions for two- dimensional non-stationary Stokes equation.

A reduced-order finite volume element formulation based on POD method and implementation of its extrapolation algorithm for unsaturated soil water flow equation

A proper orthogonal decomposition(POD) method is used to treat a classical finite volume element(FVE) formulation for unsaturated soil water flow problem.A reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional unsaturated soil water flow problem is established and the error estimates between the reduced-order FVE solutions and the classical FVE solutions and the implementation of extrapolation algorithm for solving the reduced-order FVE formulation are provided.Some numerical examples illustrate that the results of numerical computation are consistent with theoretical conclusions.Moreover,it is shown that the reduced-order FVE formulation based on POD method is feasible and efficient for solving two-dimensional unsaturated soil water flow problem.

A finite volume element formulation and error analysis for the non-stationary conduction–convection problem

The non-stationary conduction–convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element (FVE) method. A fully discrete FVE formulation and the error estimates between the fully discrete FVE solutions and the accuracy solution are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the FVE formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non-stationary conduction–convection problem.

Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations

We propose two algorithms of two-level methods for resolving the nonlinearity in the stabilized finite volume approximation of the Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. A macroelement condition is introduced for constructing the local stabilized finite volume element formulation. Moreover the two-level methods consist of solving a small nonlinear system on the coarse mesh and then solving a linear system on the fine mesh. The error analysis shows that the two-level stabilized finite volume element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite volume element solution solving the Navier-Stokes equations on a fine mesh for a related choice of mesh widths.

A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes

A model of sedimentation-consolidation processes in so-called clarifier-thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the Stokes system for an incompressible fluid describing the motion of the mixture. In cylindrical coordinates, and if an axially symmetric solution is assumed, the original problem reduces to two space dimensions. This poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted Sobolev spaces. A novel finite volume element method is introduced for the spatial discretization, where the velocity field and the solids concentration are discretized on two different dual meshes. The method is based on a stabilized discontinuous Galerkin formulation for the concentration field, and a multiscale stabilized pair of $\mathbb{P}_1$-$\mathbb{P}_1$ elements for velocity and pressure, respectively. Numerical experiments illustrate properties of the model and the satisfactory performance of the proposed method.

A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine a classical finite volume element (FVE) method with POD method to establish a reduced FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional viscoelastic problem with real practical applied background, and analyze the errors between the reduced POD FVE solution and the classical FVE solution so as to provide scientific theoretic basis for service applications. Some numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced FVE formulation based on POD method is feasible and efficient for solving two-dimensional viscoelastic problem.

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.

A finite volume element formulation for solution of elastic axisymmetric pressurized components

The authors have previously proposed the equivalent volume hypothesis (EVH) as part of integrity assessment of boiler tubes subjected to external erosion/corrosion. Currently, research is in progress at The University of New South Wales to develop and verify the EVH for tubes subjected to internal corrosion. This paper describes the formulation, verification and application of an axisymmetric and elastic finite volume element to internally corroded tubes. The initial computed results indicate that EVH is valid for these tubes.

Multiphase, Multicomponent Fluid Flow in Homogeneous and Heterogeneous Porous Media

The flow of several components and several phases through a porous medium is generally described by introducing macroscopic mass-balance equations under the form of generalized dispersion equations. This model raises several questions that are discussed in this paper on the basis of results obtained from the volume averaging method, coupled with pore-scale simulations of the multiphase flow. The study is limited to a binary, two-phase system, and we assume that the momentum equations can be solved independently from the diffusion/advection equations. The assumption of local-equilibrium is discussed and several length-scale and time-scale constraints are provided. A key issue concerns the impact on the dispersion tensors of the pore-scale equilibrium condition at the interface between the different phases. Our results show that this phenomenon may lead to significant variations of the dispersion coefficients with respect to passive dispersion, i. e. , dispersion without interfacial mass fluxes. Macroscopic equations are then obtained in the general case, and several local closure problems are provided that allow one to calculate the dispersion tensors and others properties, from the pore-scale geometry, velocities, and fluid characteristics. Examples of solutions of these closure problems are given in the case of two-dimensional representative unit cells. The two-phase flow equations are solved in two different ways : a boundary element technique, or a modified lattice Boltzmann approach. Solutions of the closure problems associated with the dispersion equations are then given using a finite volume element formulation of the partial differential equations. The results show the influence of velocity and saturation on the effective parameters. They emphasize the importance of geometry on the behavior of the dispersion tensor. Extension of these results to a larger-scale including the effect of heterogeneities is proposed in a limited case corresponding to the flow of one phase, the other phase being at residual saturation. A new large-scale dispersion equation is provided, which features a large-scale dispersion tensor that can be determined from the heterogeneity characteristics through a set of closure problems. Results are extended to a more general two-phase flow problem, when the large-scale two-phase flow can be assumed to be quasi-static. Indications are given on the difficulties associated with flow under large-scale dynamic conditions, with abnormal dispersion.

SUBGRID VOLUMETRIC QUADRATURE ACCURACY FOR TRANSIENT COMPRESSIBLE FLOW PREDICTIONS

A finite volume–element formulation of the Navier–Stokes equations for compressible flows is applied to the transient shock tube problem. A second-order spatial quadrature for volumetric integration is studied because of its effects on the shock wave resolution and positioning. Low quadrature order is shown to produce solution anomalies in regions with a transonic character as well as poor predictions of shock wave propagation. The second-order volumetric quadrature includes the proper upstream and downstream solution behaviour and eliminates both the transonic and shock speed errors in the transient shock tube problem. © 1997 John Wiley & Sons, Ltd.