Finite Volume Element Discretization Research Articles

The Finite Volume Element Method for Time Fractional Generalized Burgers’ Equation

In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers’ equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the L1 formula and approximating the spatial terms using FVEM. The convergence of the scheme is proven. Finally, numerical examples are provided to confirm the scheme’s validity.

A reduced order modeling approach with Petrov–Galerkin projection based on hybrid snapshot simulation in Semilinear-PDEs

This paper proposes an efficient and economic framework to generate reduced order models (ROMs) with Petrov–Galerkin projection based on hybrid snapshot simulation to extraordinarily enhance the speedup and enlighten the representation of snapshot data in Semilinear Partial Differential Equations (Semilinear-PDEs). The total snapshot simulation is divided into multiple temporal intervals with the alternations in each interval between two types of models: the full order model (FOM) and the local ROM based on Petrov–Galerkin projection (ROM-PG). During the snapshot training process, the simulation can determine the model type on-the-fly by the switch criteria introduced in Bai and Wang (2022) and continuously update the subspace by incremental singular value decomposition (iSVD) to preserve the subspace representation. In numerical experiments, we consider two typical Semilinear-PDEs to illustrate the proposed method successfully in finite volume element discretization: Semilinear-Parabolic Equation (Semilinear-Parabolics) and Semilinear-Convection–Diffusion Equation (Semilinear-ConvDiffs). It is found that the subspace updating is directly related to the POD numbers, the parameter μ (μp in Semilinear-Parabolics or μc in Semilinear-ConvDiffs). The tolerance of criteria also significantly affects the performance of computational acceleration. The quantitative analyses of the criteria and the brief discussion of stability about the framework are also performed to clarify the model switch and stability condition, respectively, due to no detailed discussion in the previous work (Bai and Wang 2022).

A priori error estimates of finite volume element method for bilinear parabolic optimal control problem

<abstract><p>In this paper, we study the finite volume element method of bilinear parabolic optimal control problem. We will use the optimize-then-discretize approach to obtain the semi-discrete finite volume element scheme for the optimal control problem. Under some reasonable assumptions, we derive the optimal order error estimates in $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. We use the backward Euler method for the discretization of time to get fully discrete finite volume element scheme for the optimal control problem, and obtain some error estimates. The approximate order for the state, costate and control variables is $ O(h^{3/2}+\triangle t) $ in the sense of $ L^2(J; L^2) $ and $ L^\infty(J; L^2) $-norm. Finally, a numerical experiment is presented to test these theoretical results.</p></abstract>

Parallel domain decomposition schemes based on finite volume element discretization for nonsteady-state diffusion equations on distorted meshes

In this paper, several domain decomposition algorithms are considered to solve parallelly non-steady diffusion equations. The contributions of this work are threefold. The first one is that we present effective parallel domain decomposition schemes underlying the classical and monotone finite volume element discretization. These schemes adopt a three-stage technique, which is also used in the difference method and the finite volume method. Positivity of the parallel scheme based on monotone finite volume element method is proved. The second one is that we propose a new domain decomposition computational strategy to improve the correction step in prediction-correction frame such that the whole parallel process is simplified. The third one is that we consider a parallel algorithm build on quadratic finite volume element method, which preserves local conservation and has higher convergence rates. These parallel schemes are of intrinsic parallelism since we apply the prediction-correction technique to the interface of subdomains coming from an artificial division of computational domain or multi-medium physical domain. The proposed parallel schemes need only local communication among neighboring processors. Numerical results are presented to illuminate the accuracy, stability and parallelism of the parallel schemes.

Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids

In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical -formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in -norm are derived, but it is difficult to obtain the corresponding results in -norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness.

The finite volume element method for the two-dimensional space-fractional convection\u2013diffusion equation

We develop a fully discrete finite volume element scheme of the two-dimensional space-fractional convection–diffusion equation using the finite volume element method to discretize the space-fractional derivative and Crank–Nicholson scheme for time discretization. We also analyze and prove the stability and convergence of the given scheme. Finally, we validate our theoretical analysis by data from three examples.

Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations

In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I_{h}^{*}, a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L^{infty }(L^{2}(varOmega)) and L^{2}(H^{1}(varOmega)) norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness.

Conditioning of the finite volume element method for diffusion problems with general simplicial meshes

The conditioning of the linear finite volume element discretization for general diffusion equations is studied on arbitrary simplicial meshes. The condition number is defined as the ratio of the maximal singular value of the stiffness matrix to the minimal eigenvalue of its symmetric part. This definition is motivated by the fact that the convergence rate of the generalized minimal residual method for the corresponding linear systems is determined by the ratio. An upper bound for the ratio is established by developing an upper bound for the maximal singular value and a lower bound for the minimal eigenvalue of the symmetric part. It is shown that the bound depends on three factors, the number of the elements in the mesh, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity measured in the metric specified by the inverse diffusion matrix. It is also shown that the diagonal scaling can effectively eliminates the effects from the mesh nonuniformity measured in the Euclidean metric. Numerical results for a selection of examples in one, two, and three dimensions are presented.

Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

This paper is concerned with establishing a reduced-order extrapolating finite volume element (FVE) format based on proper orthogonal decomposition (POD) for two-dimensional (2D) hyperbolic equations. For this purpose, a semi discrete variational format relative time and a fully discrete FVE format for the 2D hyperbolic equations are built, and a set of snapshots from the very few FVE solutions are extracted on the first very short time interval. Then, the POD basis from the snapshots is formulated, and the reduced-order POD extrapolating FVE format containing very few degrees of freedom but holding sufficiently high accuracy is built. Next, the error estimates of the reduced-order solutions and the algorithm procedure for solving the reduced-order format are furnished. Finally, a numerical example is shown to confirm the correctness of theoretical conclusions. This means that the format is efficient and feasible to solve the 2D hyperbolic equations.

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

We present a novel discretization scheme tailored to a class of multiphase models that regard the physical system as consisting of multiple interacting continua. In the framework of mixture theory, we consider a general mathematical model that entails solving a system of mass and momentum equations for both the mixture and one of the phases. The model results in a strongly coupled and nonlinear system of partial differential equations that are written in terms of phase and mixture (barycentric) velocities, phase pressure, and saturation. We construct an accurate, robust and reliable hybrid method that combines a mixed finite element discretization of the momentum equations with a primal discontinuous finite volume-element discretization of the mass (or transport) equations. The scheme is devised for unstructured meshes and relies on mixed Brezzi–Douglas–Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, as well as on a primal formulation that employs discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar fields of interest (such as volume fraction, total density, saturation, etc.). As the discretization scheme is derived for a general formulation of multicontinuum physical systems, it can be readily applied to a large class of simplified multiphase models; on the other, the approach can be seen as a generalization of these models that are commonly encountered in the literature and employed when the latter are not sufficiently accurate. An extensive set of numerical test cases involving two- and three-dimensional porous media are presented to demonstrate the accuracy of the method (displaying an optimal convergence rate), the physics-preserving properties of the mixed–primal scheme, as well as the robustness of the method (which is successfully used to simulate diverse physical phenomena such as density fingering, Terzaghi's consolidation, deformation of a cantilever bracket, and Boycott effects). The applicability of the method is not limited to flow in porous media, but can also be employed to describe many other physical systems governed by a similar set of equations, including e.g. multi-component materials.

Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model

Time delay is an essential ingredient of spatio-temporal predator–prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator–prey reaction–diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction–diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge–Kutta scheme.

Additive average Schwarz method for a Crouzeix–Raviart finite volume element discretization of elliptic problems with heterogeneous coefficients

In this paper we introduce an additive Schwarz method for a Crouzeix---Raviart finite volume element discretization of a second order elliptic problem with discontinuous coefficients, where the discontinuities are both inside the subdomains and across and along the subdomain boundaries. We show that, depending on the distribution of the coefficient in the model problem, the parameters describing the generalized minimal residual method (GMRES) convergence rate of the proposed method depend linearly on the mesh parameters [InlineEquation not available: see fulltext.]. Also, under certain restrictions on the distribution of the coefficient, the convergence of the GMRES method is independent of jumps in the coefficient.

Additive Schwarz preconditioner for the finite volume element discretization of symmetric elliptic problems

A symmetric and a nonsymmetric variant of the additive Schwarz pre- conditioner are proposed for the solution of a class of finite volume element discretization of the symmetric elliptic problem in two dimensions, with large jumps intheentriesofthecoefficientmatricesacrosssubdomains.Itisshownthattheconver-

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

The sedimentation–consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low Reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the Stokes problem written in terms of the mixture flow velocity and the pressure field. Here both the viscosity and the forcing term depend on the local solids concentration. A semi-discrete discontinuous finite volume element (DFVE) scheme is proposed for this model. The numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field, whereas the pressure is approximated by piecewise constant elements. The unique solvability of both the nonlinear continuous problem and the semi-discrete DFVE scheme is discussed, and optimal convergence estimates in several spatial norms are derived. Properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples, including flows under gravity with changing direction, a secondary settling tank in an axisymmetric setting, and batch sedimentation in a tilted cylindrical vessel.

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 New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.

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.

Convergence analysis and numerical simulation for a hyperbolic system

In this paper, we develop and analyze a finite volume element method for the approximation of the hyperbolic type equations arising in mathematical physics in one space dimension. In recent years there has been increasing attention in the literatures to the development, analysis and implementation of the solution of hyperbolic equations. In this work, we construct the discrete finite volume element schemes, and give the optimal error estimates between the finite volume element solution and the exact solution. Moreover, numerical experiments are also provided to demonstrate the behavior of the methods and confirm the efficiency and robustness of the presented approach and method. Finally, some conclusions are obtained and our further research work is proposed.

Biquadratic finite volume element method based on optimal stress points for second order hyperbolic equations

AbstractBased on optimal stress points, we develop a full discrete finite volume element scheme for second order hyperbolic equations using the biquadratic elements. The optimal order error estimates in L∞(H1), L∞(L2) norms are derived, in addition, the superconvergence of numerical gradients at optimal stress points is also discussed. Numerical results confirm the theoretical order of convergence. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Analysis of a finite volume element method for the Stokes problem

In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries. Finally, some numerical experiments that confirm the predicted behavior of the method are provided.