Weighted Quadrature Rules for Finite Element Methods

We discuss the numerical integration of polynomials times exponential weighting functions arising from multiscale finite element computations. The new rules are more accurate than standard quadratures and are better suited to existing codes than formulas computed by symbolic integration. We test our approach in a multiscale finite element method for the 2D reaction-diffusion equation. Standard finite elements usually fail to accurately solve equations with multiscale behavior. This can happen if coefficients are oscillatory or if a small parameter multiplies some of the terms in the equation. A strategy to overcome these difficulties is to use special spaces instead of the space of piecewise polynomial functions [1, 2]. However, for polynomial basis functions, standard quadratures are exact and this is not the case for more complicated spaces. We investigate quadratures to integrate elementwise products of polynomials and exponential basis functions. Such integrals appear when developing enriched methods for reaction-advection-diffusion equations [2], but also in other contexts [3]. Quadrature formulas are simpler to implement into existing finite element codes than results of symbolic integrations. We define an N -point weighted quadrature in [a, b] with weighting function w by a set of integration weights Al and integration points xl ∈ [a, b] such that ∫ b