Finite volume element method for analysis of unsteady reaction-diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.

An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

A FINITE VOLUME ELEMENT METHOD FOR THERMAL CONVECTION PROBLEMS

Consider the finite volume element method for the thermal convection problem with the infinite Prandtl number. The author uses a conforming piecewise linear function on a fine triangulation for velocity and temperature, and a piecewise constant function on a coarse triangulation for pressure. For general triangulation the optimal order H1 norm error estimates are given.

Finite Volume Element Predictor-corrector Method for a Class of Nonlinear Parabolic Systems

A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.

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 fi- nite 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 for- mat 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.

Biquartic Finite Volume Element Method Based on Lobatto-Guass Structure

In this paper, a biquartic finite volume element method based on Lobatto- Guass structure is presented for variable coefficient elliptic equation on rectangular partition. Not only the optimal H1 and L2 error estimates but also some superconvergent properties are available and could be proved for this method. The numerical results obtained by this finite volume element scheme confirm the validity of the theoretical analysis and the effectiveness of this method.

Quadratic Finite Volume Element Schemes over Triangular Meshes for a Nonlinear Time-Fractional Rayleigh-Stokes Problem

In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.

A STABILIZED CRANK-NICOLSON MIXED FINITE VOLUME ELEMENT FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER-STOKES EQUATIONS

A time semi-discrete Crank-Nicolson （CN） formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element （SCNMFVE） formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.

An ADI Finite Volume Element Method for a Viscous Wave Equation with Variable Coefficients

Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.

CONVERGENCE ANALYSIS OF MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT FOR THREE-DIMENSIONAL CHEMICAL OIL-RECOVERY SEEPAGE COUPLED PROBLEM

The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in 12 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.

Finite Volume Element Method for Solving the Elliptic Neumann Boundary Control Problems

Solving optimization problems with partial differential equations constraints is one of the most challenging problems in the context of industrial applications. In this paper, we study the finite volume element method for solving the elliptic Neumann boundary control problems. The variational discretization approach is used to deal with the control. Numerical results demonstrate that the proposed method for control is second-order accuracy in the L2 (Γ) and L∞ (Γ) norm. For state and adjoint state, optimal convergence order in the L2 (Ω) and H1 (Ω) can also be obtained.

Maximum Norm Estimates for Finite Volume Element Method for Non-selfadjoint and Indefinite Elliptic Problems

In this paper, we establish the maximum norm estimates of the solutions of the finite volume element method （FVE） based on the P1 conforming element for the non-selfadjoint and indefinite elliptic problems.

Uniform Convergence for Finite Volume Element Method for Non-selfadjoint and Indefinite Elliptic Problems

In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.

Derivations of Vector Area and Volume Elements in Curved Coordinate Systems for Flux Vector Fields Helping Eye Disease

Our aim in this paper is to interest retinal eye specialists in preventing dry macula degeneration by a special flurry vector field through open or closed curved surfaces. The flux of vector fields through surfaces is based on vector element area and volume element. Therefore, we explain a few geometrical derivations of area and volume elements in curved orthogonal coordinate systems. We hope that by derivation of a spatial vector field flurry against drusen through open or closed surfaces due to the Gauss theorem might select drusen under eye retina cells without destroying the cells and prevent macula degeneration. A changed flurry of a magnetic or electric vector field through a closed line causes an electric or magnetic vector field on the surface closed by the line. We also hope that derivation by Stokes’ and Greens’ theorems, with the help of iron, might help eye cells to get in life.

Mixed Finite Volume Element Method for Vibration Equations of Beam with Structural Damping

In this paper, for the initial and boundary value problem of beams with structural damping, by introducing intermediate variables, the original fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed. Numerical examples are provided to confirm the theoretical results. In the end, we test the value of δ to observe its influence on the model.

A Coercivity Result of Quadratic Finite Volume Element Schemes over Triangular Meshes

In this work,we study the coercivity of a family of quadratic finite volume element(FVE)schemes over triangular meshes for solving elliptic boundary value problems.The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as H^(1) and L^(2) error estimates.The novelty of this paper is that,each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts:the first part is the element stiffness matrix of the standard quadratic finite element method(FEM),the second part is the difference between the FVE and FEM on the element boundary,while the third part can be expressed as the tensor product of two vectors.As a result,we reach a sufficient condition to guarantee the existence,uniqueness and coercivity result of the FVE solution on general triangular meshes.Moreover,based on this sufficient condition,some minimum angle conditions with simple,analytic and computable expressions are obtained.By comparison,the existing minimum angle conditions were obtained numerically from a computer program.Theoretical findings are conformed with the numerical results.

Mesh Conditions of the Preserving-Maximum-Principle Linear Finite Volume Element Method for Anisotropic Diffusion-Convection-Reaction Equations

We develop mesh conditions for linear finite volume element approximations of anisotropic diffusionconvectionreaction problems to satisfy the discrete maximum principle.We obtain the sufficient conditions to gurantee the both upper and lower bounds of the numerical solution when each angle of arbitrary triangle is O(∥q∥_∞h+∥g∥_∞h~2)-acute and h is small enough,where h denotes the mesh size,q and g are coefficients of the convection and reaction terms,respectively.To deal with the convection-dominated problems,we use the upwind triangle technique.For such scheme,the mesh condition can be sharper to O(∥g∥_∞h~2)-acute.Some numerical examples are presented to demonstrate the theoretical results.

A Quadratic Serendipity Finite Volume Element Method on Arbitrary Convex Polygonal Meshes

Based on the idea of serendipity element,we construct and analyze the first quadratic serendipity finite volume element method for arbitrary convex polygonalmeshes in this article.The explicit construction of quadratic serendipity element shape function is introduced from the linear generalized barycentric coordinates,and the quadratic serendipity element function space based on Wachspress coordinate is selected as the trial function space.Moreover,we construct a family of unified dual partitions for arbitrary convex polygonal meshes,which is crucial to finite volume element scheme,and propose a quadratic serendipity polygonal finite volume element method with fewer degrees of freedom.Finally,under certain geometric assumption conditions,the optimal H1 error estimate for the quadratic serendipity polygonal finite volume element scheme is obtained,and verified by numerical experiments.