New embedded boundary-type quadrature formulas for the simplex

Abstract

In this paper we consider the problem of the approximation of the integral of a smooth enough function f(x,y) on the standard simplex $\Delta _{2} \subset \mathbb{R}^{2}$ by cubature formulas of the following kind:

$${\int\limits_{\Delta _{2} } {f{\left( {x,y} \right)}dxdy} } = {\sum\limits_{\alpha = 1}^3 {{\sum\limits_{i,j} {A_{{\alpha ij}} \frac{{\alpha ^{{i + j}} }}{{\alpha x^{i} \alpha y^{j} }}f{\left( {x_{\alpha } ,y_{\alpha } } \right)} + E{\left( f \right)}} }} }$$

where the nodes $\left( x_{\alpha},y_{\alpha}\right) ,\alpha=1,2,3$ are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples.