Efficient Fast Multipole Accelerated Boundary Elements via Recursive Computation of Multipole Expansions of Integrals.

In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via integration over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. The integrals are often computed via numerical or analytical quadrature. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - they are only needed in terms of their contribution to the multipole expansion coefficients. Furthermore, the two parts of this resulting algorithm - the integration and the FMM matrix vector product - are both approximate, and their errors have to be matched to avoid wasteful computations, or poorly controlled error. We propose a new fast method for generation of multipole expansion coefficients for the fields produced by the integration of the single and double layer potentials on surface triangles; charge distributions over line segments; and regular functions over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on recursive computations of the multipole moments for $O(1)$ cost per moment with a low asymptotic constant. The method is developed for the Laplace Green's function in ${\mathbb R}^3$. The derived recursions are tested both for accuracy and performance.