--ZusammenfassungAnisotropic Interpolation with Applications to the Finite Element Method. The usual Bramble-Hilbert theory is extended for proving more refined estimates of the interpolation error. For a large class of finite elements, it is shown that one can derive benefit from the presence of small and even large angles of the elements. For bilinear shape functions on rectangular grids it is proved that interpolation and finite element approximation error coincide. As an example, we consider the finite element approximation for problems on domains containing edges. AMS Subject Classification: 65N30, 65D07

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This paper deals with the finite element displacement method for approximating isolated solutions of general quasilinear elliptic systems. Under minimal assumpt,ions on the structure of the continuous problems it is shown that the discrete analogues also have locally unique solutions which converge with quasi-optimal rates in L, and L,. The essential tools of the proof are a deformation argument and n technique using weighted L,-norms. Q 1. IntroductionLet .Q be a bounded region in EucLrDean space Rfl, n 2 2. We consider the quasilinear system : k'ind u = (uk)k=;,...,r such that -i. a i~f ( . , u, ~u ) + F;(.) u, ~u ) = o in Q, ~k = gk on ail,,, 2 P:(-, u, c/u) ni = c i+zi on a~, , i = l n i= 1 i = 1 where Ss, qk and 8: are given functions and {aQ,, aQ,) is a partition of the boundary aQ.Below Lp(Q)' and Hm-p(Q)r, m E N, 1 5 p 5 00, denote the usual LEBESGUE and SOBOLEV spaces of (real) vector valued functions 2, = (2)k)k=;,,,,,r on 9. The corresponding norms are rising inult,i-index notation, with the usual modification for p = 00. €ZFJ'(aQ,; l2)' is the closure in H",p(Q)* of the subspace of Cw-vector functions vanishing on aQ,. We shall use the usual summation convention for repeated indices k = 1, . . ., r. For example the L2-scalar product is written as follows :(u, w) = J U k W k d X . RThe above boundary value problem has the usual weak formulation in the EEZLBEBT

This paper deals with the linear finite element method for approximating general quasilinear parabolic problems. Under minimal assumptions on the regularity of the solution, it is shown that the discrete solutions exist and converge with quasioptimal rate in L. The essential tool of the proof is a new weighted norm technique.

The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions n = 1 and n = 2, sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in L. and in the energy norm.