The convergence of mimetic discretization for rough grids
Following the authors, the mimetic discretisation of the scalar or vector field on a some region \(\Omega\subset{\mathbb R^d}\) with a grid defined on it, is a special kind of a finite difference approximation of the field and certain of its operators such as the scalar product, the gradient, the divergence and so on. The discretisations are defined so that the discretised version of the so called divergence theorem for a scalar field \(u\) and a vector field \(\underline{w}\) \[ \int_\Omega u\;{div}(\underline{w})d\Omega+ \int_\Omega\underline{w}\;\nabla{u}d\Omega =\int_{\partial\Omega}u\;\underline{w}\;\underline{n}dS \] is exactly satisfied. The grids cosidered here are based on the decomposition of \(\Omega\) into convex cells. The mimetic discretisations depend on the kind of grid and differ strongly for scalar and vector fields. It is to be remarked here that the mimetic discretisation of the gradient is not explicitely given, but has to be computed from an equation resulting from the disretised divergence theorem. The authors give a rather complete introduction to the theory of mimetic discretisation in one and two dimensions, the two dimensional case being limited to the quadrilateral grids. Bounds of truncation errors are found. Relations between so called formal i.e. the traditional form of the finite difference discretisation and its mimetic version for basic operators are joint. The authors present as an example study of the convergence for the mimetic discretisation of a nonhomogeneous equation with the Laplace operator and zero Dirichlet conditions in one and two dimensions.
scientific article; zbMATH DE number 2142410