Domain decomposition methods for the diffusion equation with low-regularity solution

TL;DR: It is proved well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled, and if moreover the conditions hold independently of the discretization, convergence is ensured.

Abstract: We analyze matching and non-matching domain decomposition methods for the numerical approximation of the mixed diffusion equations. Special attention is paid to the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. The domain decomposition method can be non-matching in the sense that the traces of the finite elements spaces may not fit at the interface between subdomains. We prove well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled. If moreover the conditions hold independently of the discretization, convergence is ensured.