By Albert H. Schatz, Vidar Thomee, Wolfgang L. Wendland

These are the lecture notes of the seminar "Mathematische Theorie der finiten point und Randelementmethoden" prepared through the "Deutsche Mathematiker-Vereinigung" and held in Dusseldorf from 07. - 14. of June 1987. Finite aspect tools and the heavily similar boundary point equipment these days belong to the normal workouts for the computation of strategies to boundary and preliminary boundary worth difficulties of partial differential equations with many functions as e.g. in elasticity and thermoelasticity, fluid mechanics, acoustics, electromagnetics, scatter ing and diffusion. those tools additionally inspired the improvement of corresponding mathematical numerical research. i used to be more than happy undefined. Schatz and V. Thomee generously joined the journey of the seminar and never basically gave stimulating lectures but additionally spent rather a lot time for private dialogue with the entire individuals. The seminar in addition to those notes encompass 3 components: 1. An research of the Finite aspect procedure for moment Order Elliptic Boundary price difficulties via A. H. Schatz. II. On Finite components for Parabolic difficulties via V. Thomee. III. I30undary point equipment for Elliptic difficulties by means of \V. L. Wendland. the necessities for analyzing this booklet are simple wisdom in partial differential equations (including pseudo-differential operators) and in numerical research. It was once now not our purpose to give a accomplished account of the examine during this box, yet fairly to offer an creation and assessment to the 3 varied subject matters which shed a few mild on fresh research.

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14) An immediate consequence of the best approximation properties of these projections and the properties of the interpolant is the following basic error estimates. 1. 6), 3B. 17) Proof: u o. 14), Further Estimates for the L2 Projection, respectively respectively. the Error in HI and Negative Norms, Duality. 1 gives estimates for the error in the norms of the spaces in which the various projections were taken. We may now ask what the error of these projections are in other norms. HI? For example, what is the error of the L2 projection in We shall first show in a very simple manner that the order of accuracy of this error in HI is one order less than that in L 2 , which is analogous to the interpolant.

18) back to 2E. Th T. 16) follows by transforming . We leave the details to the reader. Super-approximation Property. The next result that will be proved is known as the super-approximation property. 4) deals with how well the subspace general function in H2 (0). Sh{O) We now consider the question of how well the special functions of the form approximated. where in applications function and l({x) can approximate a w{x)l({x) w{x) is any function in can be is usually a fixed h S CO). Functions of this form arise very often in the analysis of the finite element method and the super-approximation property has proved to be an invaluable tool.

If or approximation properties in interpolant in that. uI . However. for example. One can show that. replacing follows: Phu. 1. 20) says that L2 and HI similar to the it is different from the interpolant in the extreme case where to Hr in general. o· u-Phu is of order hr from H-r(O) from H-r(O) (see Figure 3). L2 H-r(O) is also to u1 hr. Hr to its Thus it Hr(O). double the order from We shall investigate an analogous. but more complicated. phenomena for the Section 4. (,- Sh,r(O» L2 projection can be by going to norms with lower H-r(O).