Quadrature rules for finite element approximations of 1D nonlocal problems

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 ...

External finite element approximations of eigenvalue problems

— The paper is devote d to the finit e element analysis of second order e Hipt ie eigenvalue problems in the case when the approximate domains Oh are not subdomains of the original domain fl a U. The considérations are restricted to piecewise linear approximations and in the case of eigenfunctions to simple eigenvalues. The optimum rates of convergence for hoth the approximate eigenvalues and t...

Generalized Gaussian Quadrature Rules in Enriched Finite Element Methods

In this paper, we present new Gaussian integration schemes for the efficient and accurate evaluation of weak form integrals that arise in enriched finite element methods. For discontinuous functions we present an algorithm for the construction of Gauss-like quadrature rules over arbitrarily-shaped elements without partitioning. In case of singular integrands, we introduce a new polar transforma...

Error Estimates for Finite Element Approximations of Elliptic Control Problems

We investigate finite element approximations of one-dimensional elliptic control problems. For semidiscretizations and full discretizations with piecewise constant controls we derive error estimates in the maximum norm.

Quadrature Rules for Fuzzy Integrals