Semi Analytical Solution of a Nonlinear Oblique Boundary Value Problem

TL;DR

This paper introduces a semi analytical perturbation method combined with an icosahedron-based quadrature technique to solve a nonlinear oblique boundary value problem for Laplace's equation outside a sphere, demonstrating high accuracy and convergence.

Contribution

A novel semi analytical approach using perturbation series and an icosahedron mesh quadrature for nonlinear oblique boundary problems, with effective handling of Green's function singularities.

Findings

High-precision approximations of the solution are achieved.

The method converges rapidly as the perturbation parameter decreases.

Approximations improve with increasing sphere radius.

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.