Some error estimates for the finite volume element method for a parabolic problem

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work \cite{clt11} for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in $L_2$ to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric

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 provide

Barycentric interpolation and exact integration formulas for the finite volume element method

This contribution concerns with the construction of a simple and effective technology for the problem of exact integration of interpolation polynomials arising while discretizing partial differential equations by the finite volume element method on simplicial meshes. It is based on the element-wise representation of the local shape functions through barycentric coordinates (barycentric interpolation) and the introducing of classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over the geometrical shapes defined by a barycentric dual mesh. Numerical examples are presented that illustrate the validity of the technolog

New superconvergent structures developed from the finite volume element method in 1D

New superconvergent structures are introduced by the finite volume element method (FVEM), which allow us to choose the superconvergent points freely. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in Table 1. Numerical results are given to illustrate the theoretical results

The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem

AbstractIn modern numerical simulation of prospecting and exploiting oil–gas resources and environmental science, it is important to consider a numerical method for nonlinear convection-dominated diffusion problems. Based on actual conditions, such as the three-dimensional characteristics of large-scale science-engineering computation, we present a kind of characteristic finite volume element method. Some techniques, such as calculus of variations, commutating operators, the theory of prior estimates and techniques, are adopted. Suboptimal order error estimate in L2 norm and optimal order error estimate in H1 norm are derived to determine the errors for the approximate solution. Numerical results are presented to verify the performance of the scheme

A Posteriori Error Estimate for Finite Volume Element Method of the Second-Order Hyperbolic Equations

We establish a posteriori error estimate for finite volume element method of a second-order hyperbolic equation. Residual-type a posteriori error estimator is derived. The computable upper and lower bounds on the error in the H1-norm are established. Numerical experiments are provided to illustrate the performance of the proposed estimator

Numerical modelling of metal melt refining process in ladle with rotating impeller and breakwaters

The paper describes research and development of aluminium melt refining technology in a ladle with rotating impeller and breakwaters using numerical modelling of a finite volume/element method. The theoretical aspects of refining technology are outlined. The design of the numerical model is described and discussed. The differences between real process conditions and numerical model limitations are mentioned. Based on the hypothesis and the results of numerical modelling, the most appropriate setting of the numerical model is recommended. Also, the possibilities of monitoring of degassing are explained. The results of numerical modelling allow to improve the refining technology of metal melts and to control the final quality under different boundary conditions, such as rotating speed, shape and position of rotating impeller, breakwaters and intensity of inert gas blowing through the impeller.Web of Science64266465