Recently, the first author introduced a new approach for the numerical quadrature of surface integrals in the context of boundary element methods. The handling of singularities is an essential feature. It is assumed that a global parametrization P of the surface is given as a subroutine and that P is not accessible analytically. Of particular interest are parametrizations which are based on aut...

According to a result of S. N. Bernstein, «-point Chebyshev quadrature formulas, with all nodes real, do not exist when n = 8 or n ä 10. Modifications of such quadrature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We...

We investigate the approximation of weighted integrals over Rd for integrands from Sobolev spaces mixed smoothness. prove upper and lower bounds convergence rate optimal quadratures with respect to n integration nodes functions these spaces. In one-dimensional case (d=1), we obtain right quadratures. For d≥2, bound is performed by sparse-grid on step hyperbolic crosses in function domain Rd.

We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of co...

Discontinuous Galerkin (DG) methodsCompressible Navier–Stokes equations, compressible flow haveSpectral element method a long history in computational physics and engineering to approximate solutions of partial differential equations due their high-order accuracyAccuracy geometric flexibility. However, DG is not perfect there remain some issues. Concerning robustness, has undergone an extensive...