Pseudospectral three-dimensional boundary integral method applied to a nonlinear model problem from finite elasticity

This paper concerns the numerical analysis of the pseudospectral boundary integral method for the solution of linear boundary value problems for the vector Laplace equation in a three-dimensional (3D) region formed when a small ball is removed from inside the unit ball B ⊃ R³. These problems arise as Newton iterates for a certain nonlinear model problem deriving from finite elasticity, which exhibits a mechanism under which a deformed body undergoes a degenerate form of cavitation. The unit ball represents an elastic body, with the small ball representing a 'core region,' which is mapped to a cavity by the (possibly large) deformation. This deformation satisfies the (vector) Laplace equation in the body minus the core region, together with a Dirichlet condition on the outer boundary and a nonlinear Neumann condition on the inner boundary. Newton's method yields a sequence of linear problems that are reformulated as a 6 × 6 coupled system of boundary integral equations. The authors introduce a pseudospectral (discrete global Galerkin) method for this system, using the spherical harmonics as basis functions. By extending the known approximation theory for spherical polynomials it is proved that this method converges faster than any power of 1/n when the number of degrees of freedom is O((n + 1)²). As well as providing a convergence theory for this boundary value problem, the paper also contains a number of new results on spherical polynomial approximation and on the convergence of the pseudospectral method for classical integral equations of 3D potential theory on spheres.