Finite Volume Numerical Analysis for Parabolic Equation with Robin Boundary Condition

Abstract

In this paper, finite volume method on unstructured meshes is studied for a parabolic convection-diffusion problem on an open bounded set of $R^d$ ($d = 2$ or $3$) with Robin boundary condition. Upwinding approximations are adapted to treat both the convection term and Robin boundary condition. By directly getting start from the formulation of the finite volume scheme, numerical analysis is done. By using several discrete functional analysis techniques such as summation by parts, discrete norm inequality, et al, the stability and error estimates on the approximate solution are established, existence and uniqueness of the approximate solution and the 1st order temporal norm and $L^2$ and $H^1$ spacial norm convergence properties are obtained.