The hp-mortar finite-element method for the mixed elasticity and stokes problems

The motivation of this work is to apply the hp-version of the mortar finite-element method to the nearly incompressible elasticity model formulated as a mixed displacement-pressure problem as well as to Stokes equations in primal velocity-pressure variables. Within each subdomain, the local approximation is designed using div-stable hp-mixed finite elements. The displacement is computed in a mortared space, while the pressure is not subjected to any constraints across the interfaces. By a Boland-Nicolaides argument, we prove that the discrete saddle-point problem satisfies a Babu ka-Brezzi inf-sup condition. The inf-sup constant is optimal in the sense that it depends only on the local (to the subdomains) characteristics of the mixed finite elements and, in particular, it does not increase with the total number of the subdomains. The consequences, that we are aware of, of such an important result are twofold. 1. • The numerical analysis of the approximability properties of the hp-mortar discretization for the mixed elasticity problem allows us to derive an asymptotic rate of convergence that is optimal up to “log” p in the displacement; this is addressed in the present contribution.2. • When the mortar discrete problem is inverted by substructured iterative methods based on Krylov subspaces with block preconditioners, in view of the results for conforming finite elements [1], the condition number of the solver should grow logarithmically on (p, h) and not depend on the total number of the subdomains.