Decomposition Frameworks

In this chapter, we introduce and illustrate several principles employed in the formulation of domain decomposition methods for an elliptic equation. In our discussion, we focus on a two subdomain decomposition of the domain of the elliptic equation, into overlapping or non-overlapping subdomains, and introduce the notion of a hybrid formulation of the elliptic equation. A hybrid formulation is a coupled system of elliptic equations which is equivalent to the original elliptic equation, with unknowns representing the true solution on each subdomain. Such formulations provide a natural framework for the construction of divide and conquer methods for an elliptic equation. Using a hybrid formulation, we heuristically illustrate how novel divide and conquer iterative methods, non-matching grid discretizations and heterogeneous approximations can be constructed for an elliptic equation.