The authors consider the Poisson equation \((*)\) \(-\Delta u = f(x,y)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(\Omega = (0,1) \times(0,1)\). Let \(\{t^ h_ k \}^{N_ k}_{k=0}\) be a partition of \([0,1]\), \(M_ h (0,1)\) the space of piecewise Hermite cubics on \([0,1]\) and \(M^ 0_ h(0,1) = \{v \in M_ h (0,1) : v(0) = v(1)=0\}\). Set \(V^ h : = M^ 0_ h(0,1) \otimes M^ 0_ h (0,1)\). Let \(\{\xi^ h_ l\}^{2N_ h}_{l=1}\) be the set of Gauss points in \((0,1)\) and \({\mathcal G}^ h : = \{(x,y) : x,y \in \{\xi^ h_ l\}^{2N_ h}_{l=1}\}\). Then, the piecewise Hermite bicubic orthogonal spline collocation problem to \((*)\) consists in finding \(u_ h \in V^ h\) such that \((+)\) \(-\Delta u_ h (\xi) = f(\xi)\), \(\xi \in {\mathcal G}^ h\). Problem \((+)\) has a unique solution and error estimates are known.
The authors discuss a parallel iterative domain decomposition method for solving \((+)\). The method is an additive Schwarz conjugate gradient algorithm with coarse grid and arbitrary overlap. The implementation is discussed and the results of numerical experiments are given.
For the entire collection see [Zbl 0785.00036].