Bilinear quadratures for inner products
This work addresses the need for efficient numerical integration in Galerkin methods for differential and integral equations, but the contribution appears incremental as it generalizes existing quadrature concepts.
The paper presents a method for constructing bilinear quadratures to numerically evaluate continuous bilinear maps (e.g., L² inner products) on finite-dimensional function spaces over arbitrary domains in ℝᵈ, with validation of the numerical procedure.
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.