Quadrature formulae arising from FEM-approximation
Two quadrature formulae for \(u\in C^{2s}[0,1]\) are derived. First: With \(h=1/(2n)\) holds \[ \int^ 1_ 0 u(t)dt\approx h\left[{u(0)+u(1)\over 2}+\sum^{2n-1}_{j=1}u(jh)\right]+\sum^{s- 1}_{q=1}\alpha_{qs}\sum^{2s-1}_{\ell=0 }C_{q\ell}^{(s)}[u(\ell h)+(-1)^ qu(1+\ell h)]. \] The coefficients for \(s=2,3,4\) are calculated. Second: One obtains \[ \int^ 1_ 0 u(t)dt\approx h\left\{{u(0)+u(1)\over 2}+\sum^{2n- 1}_{j=1}u(jh)+\sum^{s-1} _{q=1}[1+(-1)^ q]\alpha_{qs}\sum^ s_{\ell=-s }b_{q\ell}^{(s)}[u(\ell h)+u(1+\ell h)]\right\}. \] Also some coefficients are determined. Error bounds for both formulas are derived. It is shown that the second formula has a higher accuracy. For the error bound \(R\) of the second formula holds: For \(s=2\): \(R=0.6 h^ 4\| u^{(4)}\|_ C\), \(s=3\): \(R=1.59 h^ 6\| u^ 6\|_ C\) and \(s=4\): \(R=3.8 h^ 8\| u^{(8)}\|_ C\).
scientific article; zbMATH DE number 56223