A Study on Spline Quasi-interpolation Based Quadrature Rules for the Isogeometric Galerkin BEM

The two recently introduced quadrature schemes in [7] are investigated for regular and singular integrals, in the context of boundary integral equations arising in the isogeometric formulation of the Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand, consisting of a B-spline and of an auxiliary function, is approximated by a suitable quasi-interpolant spline. In the second scheme, the auxiliary function is approximated by applying the quasi-interpolation operator and then the product of the two resulting splines is expressed as a linear combination of particular B-splines. The two schemes are tested and compared against other standard and novel methods available in the literature to evaluate different types of integrals arising in the Galerkin formulation. When h-refinement is performed, numerical tests reveal that under reasonable assumptions, the second scheme provides the optimal order of convergence, even with a small amount of quadrature nodes. The quadrature schemes are validated also in numerical examples to solve 2D Laplace problems with Dirichlet boundary conditions.