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Bilinear quadratures for inner products. (English) Zbl 1386.65093

Summary: 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.

MSC:

65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

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References:

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