An adaptive algorithm for computing a function on the Lipschitzian boundary of a three-dimensional solid based on a prescribed gradient and its application in magnetostatics

An efficient method is developed for determining a function Φ on the Lipschitzian boundary r of a three-dimensional solid when a sufficiently smooth vector function G = ⊇Φ is prescribed and the value of Φ(x 0 ) = Φ 0 is known at a point x 0 ∈ r. In the general case, Φ is calculated by solving a nonlinear equation. The theory of monotone operators is used as a basis for proving the existence of a unique generalized solution to the equation and the convergence of finite-element approximations of the solution under natural assumptions about Φ. The proposed adaptive algorithm can be used to obtain a solution to the required accuracy by a finite-element method. The method developed in the paper is important for magnetostatics, because it ensures necessary accuracy in computing the potential induced by a coil on the boundary between different magnetic media. The algorithm was tested by computing the three-dimensional magnetic fields in several magnet systems including the spectrometer dipole magnet and the large solenoidal magnet for the L3 experiment at CERN (Geneva).