6533b7dafe1ef96bd126ea71

RESEARCH PRODUCT

Approximation von extremalflächenstücken (hyperbolischen typs) durch charakteristische räumliche vierecke

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We consider solutions z of the Cauchy-problem for hyperbolic Euler-Lagrange equations derived from a general Lagrangian f(x, y, z; zx, zy) in two independent variables x, y. z is supposed to be an extremal of the corresponding variational problem. Visualizing z as a surface in R3 we give a geometric interpretation of Lewy's well-known characteristic approximation scheme for the numerical solution of second order hyperbolic equations by approximating z via a polyhedral construction built up from subunits which consist of two characteristic triangles having one side in common but lying on different planes in R3. Utilizing ideas from Cartan-geometry one can (in an appropriate sense) introduce the “mean curvature” of these subunits and it is seen that this curvature vanishes (up to terms of higher order). That is a highly plausible result for the polyhedral approximation of “minimal surfaces”.