How to Best Choose the Outer Coarse Mesh in the Domain Decomposition Method of Bank and Jimack

In Ciaramella et al. (2020) we defined a new partition of unity for the Bank-Jimack domain decomposition method in 1D and proved that with the new partition of unity, the Bank-Jimack method is an optimal Schwarz method in 1D and thus converges in two iterations for two subdomains: it becomes a direct solver, and this independently of the outer coarse mesh one uses! In this paper, we show that the Bank-Jimack method in 2D is an optimized Schwarz method and its convergence behavior depends on the structure of the outer coarse mesh each subdomain is using. For an equally spaced coarse mesh its convergence behavior is not as good as the convergence behavior of optimized Schwarz. However, if a stretched coarse mesh is used, then the Bank-Jimack method becomes faster then optimized Schwarz with Robin or Ventcell transmission conditions. Our analysis leads to a conjecture stating that the convergence factor of the Bank-Jimack method with overlap L and m geometrically stretched outer coarse mesh cells is $$ .