Splitting a concave domain to convex subdomains

We will study the convergence property of Schwarz alternating method for concave region where the concave region is decomposed into convex subdomains. Optimality of regular preconditioner deduced from Schwarz alternating is also proved. It is shown that the convergent rate and the condition number are independent of the mesh size but dependent on the relative geometric position of subdomains. Special care is devoted to non-uniform meshes, exclusively, local properties like the shape regularity of the finite elements are utilized.