Transmission conditions with constraints in finite element domain decomposition methods for flow problems

Abstract

This work presents a conservative scheme for iteration‐by‐subdomain domain decomposition (DD) strategies applied to the finite element solution of flow problems. The DD algorithm is based on the iterative update of the boundary conditions on the interfaces between the subregions, the so‐called transmission conditions. The transmission conditions involve the essential and natural boundary conditions of the weak form of the problem, and should ensure strong continuity of the velocity and weak continuity of the traction. As a first approach, the transmission conditions are interpolated using the classical Lagrange interpolation functions. Conservation problems might arise when two adjacent subdomains have a sensibly different mesh spacing. In order to conserve any desired quantity of interest, an interface constraining is introduced: continuity of the transmission conditions are constrained under a scalar conservation equation. An example of mass conservation illustrates the algorithm.