Domain Decomposition, Operator Trigonometry, Robin Condition

The purpose of this paper is to bring to the domain decomposition community certain implications of a new operator trigonometry and of the Robin boundary condition as they pertain to domain decomposition methods and theory. In Section 2 we recall some basic facts and recent results concerning the new operator trigonometry as it applies to iterative methods. This theory reveals that the convergence rates of many important iterative methods are determined by the operator angle φ(A) of A: the maximum angle through which A may turn a vector. In Section 3 we bring domain decomposition methods into the operator trigonometric framework. In so doing a new three-way relationship between domain decomposition, operator trigonometry, and the recently developed strengthened C.B.S. constants theory, is established. In Section 4 we examine Robin–Robin boundary conditions as they are currently being used in domain decomposition interface conditions. Because the origins of Robin’s boundary condition are so little known, we also take this opportunity to enter into the record here some recently discovered historical facts concerning Robin and the boundary condition now bearing his name.