The MMOC and MMOCAA schemes for the finite volume element method of convection-diffusion problems

We present and analyze the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for the finite volume element (FVE) method of convection-diffusion problems. These two schemes maintain the advantages of both the MMOC and the FVE method. And the MMOCAA scheme discussed herein conserves the conservation law globally at a minor additional computational cost. Optimal-order error estimates in the H¹-norm are proved for these schemes. A numerical example is presented to confirm the estimates.     

A new stabilized finite volume method for the stationary Stokes equations

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P ₁–P ₁ pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H ¹-norm for velocity and an estimate in the L ²-norm for pressure are obtained. An optimal error estimate in the L ²-norm for the velocity is derived under an additional assumption on the body force. This work is supported in part by the NSF of China 10701001 and by the US National Science Foundation grant DMS-0609995 and CMG Chair Funds in Reservoir Simulation.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

Analysis of a cell-vertex finite volume method for convection-diffusion problems

A cell-vertex finite volume approximation of elliptic convection-dominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and second-order convergent in a mesh-dependent -norm.

     

An unstructured finite volume time domain method for structural dynamics

An unstructured finite volume time domain method (UFVTDM) is proposed to simulate stress wave propagation, in which the original variables of displacement and stress are solved based on the dynamic equilibrium equations. An Euler explicit and unstructured finite volume method is used for time dependent and spacial terms respectively. The displacements are stored on the cell vertex and a vertex based finite volume method is formed with that integral surface and the stresses are as assumed to be uniform in the cell. The present UFVTDM has several features. (1) The governing equations are discretized with the finite volume method which naturally follows conservation laws. (2) It can handle complex engineering problem. (3) This method is also able to analyze the natural characteristics and the numerical experiment shows that it is very efficient. Several cases are used to show the capability of the algorithm.     

Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study

The accuracy of a finite element numerical approximation of the solution of a partial differential equation can be spoiled significantly by singularities. This phenomenon is especially critical for high order methods. In this paper, we show that, if the PDE is linear and the singular basis functions are homogeneous solutions of the PDE, the augmentation of the trial function space for the Finite Volume Element Method (FVEM) can be done significantly simpler than for the Finite Element Method. When the trial function space is augmented for the FVEM, all the entries in the matrix originating from the singular basis functions in the discrete form of the PDE are zero, and the singular basis functions only appear in the boundary conditions. That is to say, there is no need to integrate the singular basis functions over the elements and the sparsity of the matrix is preserved without special care. FVEM numerical convergence studies on two-dimensional triangular grids are presented using basis functions of arbitrary high order, confirming the same order of convergence for singular solutions as for smooth solutions.     

Two-time level ADI finite volume method for a class of second-order hyperbolic problems

In this paper, we develop a two-time level alternating direction implicit (ADI) method for a class of second-order hyperbolic problems on a rectangular domain. The method builds on the finite volume method with biquadratic basis functions for the discretization in space, and a Crank-Nicolson approach for the time stepping. We obtain a second-order error estimation in the H¹ norm. Numerical experiments are performed to demonstrate the theoretical findings.     

A posteriori error estimator for finite volume methods

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.     

A projection-based stabilized finite element method for steady-state natural convection problem

We formulate a projection-based stabilization finite element technique for solving steady-state natural convection problems. In particular, we consider heat transport through combined solid and fluid media. This stabilization does not act on the large flow structures. Based on the projection stabilization idea, finite element error analysis of the problem is investigated and optimal errors for the velocity, temperature and pressure are established. We also present some numerical tests which both verify the theoretical predictions and demonstrate the method?s promise.     

A multilevel finite volume method with multiscale-based grid adaptation for steady compressible flows

An implicit multilevel finite volume solver on adaptively refined quadtree meshes is presented for the solution of steady state flow problems. The nonlinear problem arising from the implicit time discretization is solved by an adaptive FAS multigrid method. Local grid adaptation is performed by means of a multiscale-based strategy. For this purpose data of the flow field are decomposed into coarse grid information and a sequence of detail coefficients that describe the difference between two refinement levels and reveal insight into the local regularity behavior of the solution. Here wavelet techniques are employed for the multiscale analysis. The key idea of the present work is to use the transfer operators of the multiscale analysis for the prolongation and restriction operator in the FAS cycle. The efficiency of the solver is investigated by means of an inviscid 2D flow over a bump.     

Convergence analysis of a finite volume method via a new nonconforming finite element method

The article is devoted to the study of convergence properties of a Finite Volume Method (FVM) using Voronoi boxes for discretization. The approach is based on the construction of a new nonconforming Finite Element Method (FEM), such that the system of linear equations coincides completely with that for the FVM. Thus, by proving convergence properties of the FEM, we obtain similar ones of the FVM. In this article, the investigations are restricted to the Poisson equation. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:213–231, 1998     

A finite volume method on general surfaces and its error estimates

In this paper, we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. The optimal error estimates of the approximate solution are proved in both the H¹ and L² norms which are of first order and second order respectively under mesh regularity assumptions. Some numerical tests are also carried out to experimentally verify our theoretical analysis.     

Lax theorem and finite volume schemes

This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate of all TVD schemes for linear advection in 1D is an application of the general result. Using duality techniques and assuming enough regularity of the solution, convergence of the upwind finite volume scheme for linear advection on a 2D triangular mesh is proved in , : provided the solution is in , it proves a rate of convergence in .

     

Cubic superconvergent finite volume element method for one-dimensional elliptic and parabolic equations

In this paper, a cubic superconvergent finite volume element method based on optimal stress points is presented for one-dimensional elliptic and parabolic equations. For elliptic problem, it is proved that the method has optimal third order accuracy with respect to H¹ norm and fourth order accuracy with respect to L² norm. We also obtain that the scheme has fourth order superconvergence for derivatives at optimal stress points. For parabolic problem, the scheme is given and error estimate is obtained with respect to L² norm. Finally, numerical examples are provided to show the effectiveness of the method.     

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     

Uniform convergence of finite volume element method with Crouzeix–Raviart element for non-self-adjoint and indefinite elliptic problems

In this paper, we consider the finite volume element method based on the Crouzeix–Raviart element and prove the existence, uniqueness and uniform convergence of the finite volume element approximations for the non-self-adjoint and indefinite elliptic problems under minimal elliptic regularity assumption.     

Finite volume and finite element methods for solving a one-dimensional space-fractional Boussinesq equation

In this paper, a new one-dimensional space-fractional Boussinesq equation is proposed. Two novel numerical methods with a nonlocal operator (using nodal basis functions) for the space-fractional Boussinesq equation are derived. These methods are based on the finite volume and finite element methods, respectively. Finally, some numerical results using fractional Boussinesq equation with the maximally positive skewness and the maximally negative skewness are given to demonstrate the strong potential of these approaches. The novel simulation techniques provide excellent tools for practical problems. These new numerical models can be extended to two- and three-dimensional fractional space-fractional Boussinesq equations in future research where we plan to apply these new numerical models for simulating the tidal water table fluctuations in a coastal aquifer.     

A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations

In this paper, the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular section are solved using combined finite volume method and spectral element technique, improved by means of Hermit interpolation. The transverse applied magnetic field may have an arbitrary orientation relative to the section of the pipe. The velocity and induced magnetic field are studied for various values of Hartmann number, wall conductivity and orientation of the applied magnetic field. Comparisons with the exact solution and also some other numerical methods are made in the special cases where the exact solution exists. The numerical results for these sample problems compare very well to analytical results.     

Adaptive finite element analysis of structures under transient dynamic loading using modal superposition

In this paper, the error estimation and adaptive strategy developed for the linear elastodynamic problem under transient dynamic loading based on the Z–Z criterion is utilized for 2D and plate bending problems. An automatic mesh generator based on “growth meshing” is utilized effectively for adaptive mesh refinement. Optimal meshes are obtained iteratively corresponding to the prescribed domain discretization error limit and for a chosen number of basis modes satisfying modal truncation errors. Numerous examples show the effectiveness of the integrated approach in achieving the target accuracy in finite element transient dynamic analysis.