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Bilinear Quadratures and Their Applications


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. The evaluation of such maps is common in Galerkin methods for differential and integral equations. A framework for constructing bilinear quadratures over arbitrary domains in $\R^d$ is developed. We provide rigorous error estimates under general conditions. We prove that in one dimension, these integration rules include Gaussian quadrature for polynomials and the trapezoidal rule for trigonometric polynomials as special cases. Lastly, we demonstrate that in higher dimensions, bilinear quadratures computed using this framework achieve similar or better performance in evaluating $L^2$ inner products than existing quadrature rules.

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