A study on spline quasi-interpolation based quadrature rules for the isogeometric Galerkin BEM

TL;DR

This paper investigates two spline quasi-interpolation based quadrature schemes for weakly singular integrals in isogeometric Galerkin BEM, demonstrating their effectiveness and optimal convergence in numerical solutions of 2D Laplace problems.

Contribution

It introduces and compares two novel spline quasi-interpolation quadrature schemes for boundary element methods, showing their optimal convergence with minimal quadrature nodes.

Findings

Second scheme converges optimally with h-refinement.

Both schemes perform well compared to standard methods.

Validated on 2D Laplace problems with Dirichlet boundary conditions.

Abstract

Two recently introduced quadrature schemes for weakly singular integrals [Calabr\`o et al. J. Comput. Appl. Math. 2018] are investigated in the context of boundary integral equations arising in the isogeometric formulation of Galerkin Boundary Element Method (BEM). In the first scheme, the regular part of the integrand is approximated by a suitable quasi--interpolation spline. In the second scheme the regular part is approximated by a product of two spline functions. The two schemes are tested and compared against other standard and novel methods available in literature to evaluate different types of integrals arising in the Galerkin formulation. Numerical tests reveal that under reasonable assumptions the second scheme convergences with the optimal order in the Galerkin method, when performing $h$-refinement, 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.