A PSEUDOSPECTRAL 3-DIMENSIONAL BOUNDARY INTEGRAL METHOD APPLIED TO A NONLINEAR MODEL PROBLEM FROM FINITE ELASTICITY

This paper concerns the numerical analysis of the pseudospectral bound ary integral method for the solution of linear boundary value problems for the vector Laplace equation in a three-dimensional (3D) region fo rmed when a small ball is removed from inside the unit ball B subset o f R(3). These problems arise as Newton iterates for a certain nonlinea r model problem deriving from finite elasticity, which exhibits a mech anism under which a deformed body undergoes a degenerate farm of cavit ation. The unit ball represents an elastic body, with the small ball r epresenting a ''core region,'' which is mapped to a cavity by the (pos sibly large) deformation. This deformation satisfies the (vector) Lapl ace equation in the body minus the core region, together with a Dirich let condition on the outer boundary and a nonlinear Neumann condition on the inner boundary. Newton's method yields a sequence of linear pro blems that are reformulated as a 6 x 6 coupled system of boundary inte gral equations. The authors introduce a pseudospectral (discrete globa l Galerkin) method for this system, using the spherical harmonics as b asis functions. By extending the known approximation theory for spheri cal polynomials it is proved that this method converges faster than an y 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 pro blem, the paper also contains a number of new results on spherical pol ynomial approximation and on the convergence of the pseudospectral met hod for classical integral equations of 3D potential theory on spheres .