Well-posedness and finite volume approximation of elastic-elastic interaction model with jump embedded boundary conditions

Abstract

This paper addresses an elastic--elastic interaction problem governed by jump-embedded boundary conditions (J.E.B.C.) on the interface between two subdomains, based on the fictitious domain principle. We develop a variational formulation and prove the existence and uniqueness of weak solutions using the Lax--Milgram theorem. A penalized problem is introduced to approximate spring-law-type interface conditions, and strong convergence is established, with error estimates explicitly depending on the positive penalization parameter. A finite volume discretization is proposed, preserving local conservation in interior control volumes (CVs), while treating interface CVs separately due to the jump conditions. Numerical experiments demonstrate convergence rates between first and second order under various interface conditions, confirming the theoretical results. The proposed framework provides a reliable foundation for further developments in elasticity interface modeling.