Cubature of integrands containing derivatives
The authors present an accurate quadrature method for approximation to planar integrals \[ \iint_A(\nabla u)^Ta(\nabla v)dx dy\tag{i} \] useful in finite element approximations. \(A\) is either a parallelogram or triangle bounded by vectors \(I_1\), \(I_2\). It is sufficient to approximate integrals \[ \iint_A(\partial u/\partial n_i)a(\partial v/\partial n_j)dx dy,\quad i,j= 1,2,\tag{ii} \] \(n_1= I_1/| I_1|\), \(n_2= I_2/| I_2|\). For the parallelogram, \(A\) is partitioned into \((m+ 1)^2\) subdivisions similar to \(A\) and (ii) is approximated by a Riemann sum \(J^{(m)}\), where \(u\), \(v\) are evaluated at vertex points and \(a\) at center points of the grid. For \(u,v,a\in C^{p+ 1}\) in an open region about \(A\) it is proved that the error in applying \(J^{(m)}\) has the form \[ \sum^{p- 1}_{q= 1} D_q/m^q+ O(1/m^p).\tag{iii} \] The integrals (ii) are approximated by extrapolating values of \(J^{(m)}\), \(m= 1,2,4,8,\dots\), as in Romberg quadrature. The authors present a similar discretization when \(A\) is a triangle. Results are presented for a specific example where \(A\) is the triangle bounded by \((0,0)\), \((1,0)\), \((1,1)\). 15-significant figure accuracy was obtained.
scientific article; zbMATH DE number 1111584
scientific article