Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. (English) Zbl 0909.35154

Summary: Bernoulli’s free-boundary problem arises in ideal fluid dynamics, optimal insulation and electro chemistry. In electrostatic terms we want to design an annular condenser with a prescribed and an unknown boundary component such that the electrostatic field is constant in magnitude along the free boundary. Typically the interior Bernoulli problem has two solutions, an elliptic one close to the fixed boundary and a hyperbolic one far from it. Previous results mainly deal with elliptic solutions exploiting their monotonicity as discovered by A. Beurling. Hyperbolic solutions are more delicate for analysis and numerical approximation. Nevertheless, we derive a second-order trial free-boundary method, the implicit Neumann scheme, with equally good performance for both types of solutions. Super linear convergence of a semi-discrete variant is proved under a natural non-degeneracy condition. Numerical examples computed by this method confirm analytic predictions including questions of uniqueness, connectedness, elliptic and hyperbolic limits.


35R35 Free boundary problems for PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
65Z05 Applications to the sciences
76B99 Incompressible inviscid fluids
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