Semi Analytical Solution of a Nonlinear Oblique Boundary Value Problem
Journal:
arXiv
Published Date:
Jun 27, 2025
Abstract
A new numerical method is developed to approximate the solution of Laplace's
equation in the exterior of the sphere with a strongly nonlinear boundary value
of oblique type. A functional analysis attempt to solve this type of boundary
condition is not straight forward since results about existence and uniqueness
of solution are still limited. Hence, a semi analytical method is described
here to approach a solution. A perturbation solution around the monopole
converts the nonlinear oblique problem into a series of known Neumann problems
in the exterior of the sphere. The corresponding Green's function
representation for the exterior Neumann problem gives an exact analytic
solution for each perturbation step as an integral on the surface of the
sphere. Nevertheless, the boundary conditions become very complicated and
require to be approximated numerically. The perturbation solutions given by
integrals of the Green's function on the sphere are computed at each
perturbation step using different subdivisions of the surface integrals with
the help of adaptive quadrature method. We call icosahedron method to the
integration on the sphere with an icosahedron mesh using Gauss 5-point or
adaptive quadrature, according to the integration parameter. This method was
very effective to deal with the singularity of the Green's function
successfully avoiding inaccuracies on the numerical approximation and is an
important contribution of this work. The numerical perturbation scheme is
performed for two given exact solutions. The icosahedron method is found to be
very precise. The approximations show the desired properties: they get closer
to the exact solutions as the perturbation parameter gets smaller, show rapid
convergence in the exterior of the unit sphere and converge to zero as the
radius grows.
