We examine a cell-vertex finite volume method which is applied to a model parabolic convection-diffusion problem. By using techniques from finite element analysis, local errors away from all layers are obtained in a seminorm that is related to, but weaker than, the L2 norm.

In this paper, we present an overview of the progress of the finite volume element (FVE) methods. We show that the linear FVE methods are quite mature due to their close relationship to the linear finite element methods, while development of higher order finite volume methods remains a difficult and promising research front. Theoretical analysis, as well as the algorithms and applications of th...

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 [Math. Comp. 81 (2012), 1–20] for the lumped mass method carry over to the present situation. In particular, in order for er...

Practical engineering problems in heat transfer and fluid flow involve one or more governing equations, together with some boundary conditions over a domain. The domain is often complex and non-uniform. The ability to use a mesh of finite elements to accurately discretise domain of any size and shape makes the finite element method a powerful tool to numerically analyse problems in these areas....

The occurrence of unphysical negative concentrations in solutions to diffusion equations is a well known and severe problem. Especially in three dimensions finite element discretizations give qualitatively very poor results, while finite volume discretizations are much more stable. We investigate the cause of these instabilities and trace them back to constraints on the mesh. It turns out that ...

A general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed. Basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated. Some numerical results are given to show the effectiveness and efficiency of the...

In this study, a mathematical model is introduced to simulate the coupled heat transfer equation and Stefan condition occurring in moving boundary problems such as the solidification process in the continuous casting machines. In the continuous casting process, there exists a two-phase Stefan problem with moving boundary. The control-volume finite difference approach together with the boundary ...

A hybrid method to simulate unsteady multiphase flows is described. One phase is represented by moving particles and the other phase is defined on stationary mesh. The flow field is discretized by a conservative finite volume approximation on the stationary mesh, and the interface is automatically captured by the distribution of particles moving through the stationary mesh. The effects of surfa...

The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) “Mixed” approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through local meshless interpolation schemes such as the moving least squares(MLS) or radial basis fun...