Exterior axisymmetric Neumann problem for the Laplace equation: Numerical algorithms without saturation

Abstract

The development of conceptually new—unsaturated—numerical methods is closely related to Babenko’s work [1]. A distinctive feature of these methods is that they lack the principal error term. As a result, they are able to adjust automatically to any natural correctness classes of problems [2]. Babenko’s fundamental ideas are used to develop an unsaturated method for the numerical solution of an external axisymmetric Neumann problem for the Laplace equation. In the case of C ∞ -smooth solutions, the method has an absolutely sharp exponential error estimate up to a slowly growing factor. This result is fundamental, since the sharpness of the error estimate is determined by the asymptotics of the Aleksandrov width of the compact set of C ∞ -smooth functions containing the desired solution. This asymptotics also has the form a decaying exponential function [3]. The method is validated by computing the nonseparated axisymmetric potential flow of an inviscid incompressible fluid past an ellipsoid with an aspect ratio of 1000 (see [4]). Note that any saturated methods (i.e., with a principal error term), such as finite differences, finite elements, the quadrature method, etc., fail to produce a satisfactory solution to the problem even for an aspect ratio of 25. Success in the solution of this axisymmetric problem is reached solely due to achievements in the theory of constructive function approximation on the interval. 1. Let x ∈ � 3 , x = ( x , y , z ) , and ω ⊂ � 3 be a domain with an axis of symmetry z bounded by a smooth closed surface of revolution ∂ω . The meridional section of ∂ω is the curve γ : [0, π ] → { r ( s ), z ( s )}, γ ( s ) ∈ C ∞ [0, π ] . Then γ (0) and γ ( π ) are the poles of ∂ω . The values r = and z are invariant under the group of rotax 2 y 2 + tions of ω about the z axis. The normal N on ∂ω is assumed to be directed inward � 3 \ ω . The locations of the points x = ( r , z ) and ξ = ( ρ , ζ ) on γ are defined by coordinates s and σ , respectively: r = r ( s ), z = z ( s ); ρ = r ( σ ), ζ = z ( σ ) . If x ∈ ∂ω , then N ≡ N ( x ) = N ( s ) . The direct value of F ( x ) on ∂ω (if any) is denoted by ( x ) = F ( x ) | x ∈ ∂ω . The limiting value of the derivative = ∇ x F 〈 N 〉 from the outside of ∂ω and its direct value on ∂ω (if they exist) are designated as ( N – )( x ) and ( N )( x ), respectively. The exterior Neumann problem is to find a function Φ ( x ) that obeys the conditions