Finite volume methods with local refinement for convection-diffusion problems
This paper deals with the construction of conservative cell-centered approximations on locally refined grids for convection-diffusion second- order elliptic equations that have optimal order of convergence and satisfy the discrete maximum principle. In order to produce monotone schemes for convection-diffusion problems a variety of upwinding strategies have been used for a long time. In this paper, while keeping the required regularity of the solution to \(H^ s\) for \(3/2\leq s\leq 3\), the authors derive monotone approximations on grids with local patch refinement, estimate the error in the natural discrete \(H^ 1\)-norm and prove convergence rates of \(O(h^{1/2})\), or \(O(h^{3/2})\) depending upon the interpolation (piecewise constant or linear) used and also depending if the simple or modified upwinding strategies are used. Computer experiments for a variety of convection-diffusion problems, including convection dominated ones, in support of theoretical results and to assess the applicability of the derived schemes and error bounds are presented.
conservative cell-centered approximations
scientific article; zbMATH DE number 647753
scientific article