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Such estimates may be applied to include an adaptive aspect in the solution method, by detecting areas in a computational domain where the error is larger than elsewhere, and using this information to reÿne the mesh locally to reduce the error by an additional computation. Pioneering work is BabuÄska (1976) and BabuÄska and Rheinboldt (1978); for a recent survey, see Verfurth [40]. 1) in a polygonal domain , using piecewise linear ÿnite element approximations. With { j } the basis of pyramid functions we deÿne j by j = supp j .

In some situations one may want to use ÿnite element spaces Sh deÿned by piecewise polynomial approximating functions on a partition Th of which are not continuous across interelement boundaries, so called nonconforming elements. Assuming polygonal so that it is exactly a union of elements , one may introduce a discrete bilinear form by Dh ( ; ) = ∈Th ( ;  ) . 12) 1; h As an example, consider an axes parallel rectangular domain, partitioned into smaller such rectangles with longest edge 6h, and let Sh be piecewise quadratics which are continuous at the corners of the partition.

This method may be described as cell centered and goes back to Tikhonov and Samarskii (1961) in the case of rectangular meshes; for further developments, see Samarskii, Lazarov and Maharov [30]. For such meshes it was used in Varga [39] to construct ÿnite di erence schemes. An associated method is the following vertex centered method, also referred to as the ÿnite volume element method: Let Sh ⊂ H01 be the piecewise linear ÿnite element space deÿned by Th . The straight lines connecting a vertex of ∈ Th with the midpoint of the opposite edge intersect at the barycenter of and divide into six triangles.

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