A Two-Stage Multi-Splitting Method for Non-Overlapping Domain Decomposition for Parabolic Equations

In domain decomposition for parabolic partial differential equations (PDE) several approaches have been developed— breaking the domain into multiple subdomains of either overlapping or non-overlapping type, or using algebraic type splittings— cf. [CM94] for an overview. An important aspect is how to present the boundary conditions across interfaces or across common unknown points of subdomains, cf. [GS98, HT96, Tan92]. Towards parallelism, we divide the domain into subdomains with one grid point in common, adding an extra unknown at the interface to have effectively a non-overlapping decomposition. In the present numerical method we have designed a one gridpoint overlap together with an extra equation in order to arrive at an effective multi-splitting approach. The transmission of data at the interface is through a discrete parametrized Robin boundary condition across interior interface points. A significant part of this report is the design and experimental study of optimizing boundary parameter coupled with particular choices of inner and outer splittings. We are interested here in extending some work of San and Tang [HT96] and Tang [Tan92] to parabolic problems. There is a parameter γ that acts like a feedback gain across the artificial interfaces. The primary aspect of this article is to construct and demonstrate effective multi-splitting methods as depending on the interface boundary condition. Consider the numerical solution of parabolic problems of form: