By Heinz W. Engl, Ewald Lindner (auth.), a. Univ. Prof. Dipl.-Ing., Dr. Heinz W. Engl, o. Univ. Prof. Dr. Hansjörg Wacker, Dipl.-Ing. Dr. Walter Zulehner (eds.)

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2) is the unique minimizer of the form f(u) 1 2 a(u,u) - l(u). Proof: see [ 3] . 37 More can be said for bodies of revolution. The mapping Z: (0 1 00 ) X JR X [0,271[ -+ JR 3 describes the transformation from cylindrical coordinates to Cartesian coordinates. If G is symmetric about the x 3 -axis, it can be written in the form G z ( s x [ o, 2 n [ ) with s c. ( o,"") x JR • s G Fig. 3 The space H1 (G) can be identified by the corresponding space H~ 71 (S x [0,2n[) of functions in cylindrical coordinates.

Y and the density of steel as functions of temperature (cf. [17], [22]). Note that the density does not change much with the temperature, which justifies to assume a constant density. We now turn to the model of continuous (slab) casting we use. ByD, we denote the two-dimensional cross-section of the strand in (x,y)-space. one at t=O). The movement of the strand is modelled in such a way as if D were kept fixed and the casting machine were moved upward with casting speed v, so that the cooling zones move past the cross-section D.

4 Numerical experiments First, the numerical method is tested on a problem with known exact solution. Then a model of a turbine in a pipe filled with water is considered. All computations were performed on an IBM 360/44 system in single precision. 44 Example A: ellipsoids of revolution The surfaces 3G 2 and 3G 1 are confocal ellipsoids of revolution. The common foci are (0,0,1) and (0,0,-1). , see Fig. 5. G Fig. 5 The inner and outer boundary of S are uniformly divided into MZ parts, the number of subdivisions between inner and outer boundary is MR.