Bilinear quadratures for inner products

Abstract

A bilinear quadrature numerically evaluates a continuous bilinear map, such as the $L^2$ inner product, on continuous $f$ and $g$ belonging to known finite-dimensional function spaces. Such maps arise in Galerkin methods for differential and integral equations. The construction of bilinear quadratures over arbitrary domains in $\mathbb{R}^d$ is presented. In one dimension, integration rules of this type include Gaussian quadrature for polynomials and the trapezoidal rule for trigonometric polynomials as special cases. A numerical procedure for constructing bilinear quadratures is developed and validated.