Method of asymptotic partial decomposition with discontinuous junctions for the wave equation

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the wave equation, these conditions ensure the continuity of the solution and continuity in average of the normal flux. However, some numerical approaches prefer to have more degrees of liberty for the approximate solution. For example, the finite elements method (FEM) may use discontinuous elements. That is why the strong continuity condition for the solution may be replaced by more flexible condition of continuity in average over the interface cross section. In the present paper, we introduce and justify this alternative junction condition allowing such discontinuity of the approximate solution of the wave equation with Neumann's boundary condition. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. A finite volume scheme is used to solve the wave equation as a hybrid dimension problem. The convergence of this scheme, when the wave equation is formulated as a second-order partial differential equation in space and time, is proved using energy techniques.