On Schwarz's domain decomposition methods for elliptic boundary value problems (Q1968788)

The authors study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problems of order \(2r\) based on a spline space of smoothness \(r-1\). It is shown that the approximate solutions for the multiplicative Schwarz domain decomposition method converge geometrically to the exact solution of the linear system generated by the finite element method and that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. Numerical experiments with the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using \(C^1\) cubic spline functions over a quadrangulation of the given domain demonstrate that the greater the overlap of the subdomains the better the condition numbers and the convergence rate.