Symmetric Triangle Quadrature Rules for Arbitrary Functions

Abstract

Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this work, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.