Summary. This paper is concerned with the stability of multistep methods for ordinary initial-value problems on grids with variable mesh-sizes. A necessary and sufficient condition for stability is given from which generalizations of recent results by Gear et al. and by Zlatev can be obtained as special cases. As an application the stability of the variable BDF-formulas is treated.

In this paper we study the convergence of a centred finite difference scheme on a non-uniform mesh for a 1D elliptic problem subject to general boundary conditions. On a non-uniform mesh, the scheme is, in general, only first-order consistent. Nevertheless, we prove for s ∈ (1/2, 2] order O(h s)-convergence of solution and gradient if the exact solution is in the Sobolev space H 1+s (0, L), i.e. the so-called supraconvergence of the method. It is shown that the scheme is equivalent to a fully discrete linear finite-element method and the obtained convergence order is then a superconvergence result for the gradient. Numerical examples illustrate the performance of the method and support the convergence result.

This paper deals with the supraconvergence of elliptic finite difference schemes on variable grids for second order elliptic boundary value problems subject to Dirichlet boundary conditions in two-dimensional domains. The assumptions in this paper are less restrictive than those considered so far in the literature allowing also variable coefficients, mixed derivatives and polygonal domains. The nonequidistant grids we consider are more flexible than merely rectangular ones such that, e.g., local grid refinements are covered.The results also develop a close relation between supraconvergent finite difference schemes and piecewise linear finite element methods. It turns out that the finite difference equation is a certain nonstandard finite element scheme on triangular grids combined with a special form of quadrature. In extension to what is known for the standard finite element scheme, here also the gradient is shown to be convergent of second order, and so our result is also a superconvergence result for the underlying finite element method. o

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