Bilinear quadratures for inner products

TL;DR

This paper introduces a general method for constructing bilinear quadratures to numerically evaluate inner products on finite-dimensional function spaces, unifying and extending classical quadrature rules.

Contribution

It presents a novel numerical procedure for constructing bilinear quadratures applicable to arbitrary domains, generalizing classical rules like Gaussian quadrature and trapezoidal rule.

Findings

The procedure successfully constructs bilinear quadratures for various domains.

The method encompasses classical quadrature rules as special cases.

Numerical validation confirms the effectiveness of the constructed quadratures.

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.