Optimized Domain Decomposition Methods For The Laplace Equation On The Sphere

In this paper, we investigate various optimized domain decomposition methods solving the boundary problem of Laplace equation defined on the surface of a sphere. Domain decomposition methods, including Dirichlet-Neumann, Neumann-Dirichlet, Neumannn-Neumann method and so on, with a overlap or not are discussed for the case of two subdomains or more than two subdomains. Convergence and the finite-step convergence of the method are obtained.By the one of the coordinates transformation from Descartes coordinates to spherical coordinates, the boundary problem of Laplace equation defined on the surface of a sphere, can be solved by separate variable methods, and then the general solution can be obtained. Hence, by expanded in a Fourier series, the problem can be transformed into a family of ordinary differential equations and thereby two fundamental solutions can be constructed.In the case of two subdomains, the subdomains are decomposed by latitude. Nonoverlapping Dirichlet-Neumann alternate method, Neumann-Dirichlet alternate method, as well as waveform relaxations Neumann-Neumann method, Neumann-Dirichlet method, Dirichlet-Neumann method, are discussed. Convergence rates of these methods are estimated and two-step convergence is obtained. As for overlapping methods, by introducing Robin inner boundary transmission conditions, optimized domain decomposition method which can accelerate the convergence of classical Schwarz method are obtained. Convergence and two-step convergence of the methods are obtained. Above results can be extended to the methods with more than two subdomains.Numerical analysis is presented in the paper. In the process of discretization, the uniform grid is shifted half mesh away from the poles by using the symmetry constrain property of Fourier coefficient on the sphere. In this way, the singularities at the poles can be avoided. Based on this, the second- and fourth-order centered difference of schemes are established. Numerical results illustrate the effectiveness of the optimized domain decomposition methods we proposed compared with the classical Schwarz methods.