Rapid zonal algorithm for polyelliptic PDEs in domains with high aspect ratio

Abstract

Presented herein is a zonal boundary element method (ZBEM) for the rapid and efficient solution of a wide class of polyelliptic boundary value problems which can be recast in integral-equation form, in domains with high aspect ratio ( L⪢ 1). In contrast to the dense-matrix solution procedure of the classical BEM (CBEM), the ZBEM employs a sparse, block-tridiagonal matrix solution technique which admits rapid inversion. Our large-L asymptotic theory predicts the ZBEM to be O( L 2) times faster than, and require O( L −1) times the storage of, the equivalent-resolution CBEM. By implementing the ZBEM on two engineering-based harmonic and biharmonic example boundary value problems, up to l = 1000, we are able to demonstrate excellent agreement between our numerical results and our asymptotic theory. We suggest that the ZBEM permits the economical solution of a wide class of problems which were hitherto resolvable on only the largest computational platforms.