Coupled analysis of independently modeled finite element subdomains

The three‐field formulation for elasticity problems

“…An important choice to be made is whether to use continuous or discontinuous finite elements. Although the choice of continuous finite element approximations for the Lagrange multipliers μ k was advocated, for instance, in [1], [2], we believe that allowing discontinuities is a healthy choice, at least when the interface Σ is not smooth (which means, in this context, that its normal unit vector n has jumps that do not tend to zero with the mesh size), as it will almost always be the case when dealing with more than two subdomains. Indeed, in view of (2.35), the solution λ k has to jump whenever n jumps (as we expect each σ σ σ(u k ) to be smooth).…”

“…And it is quite reasonable to assume that u is smooth there. Sometimes, in particular for two-dimensional problems (where Σ is a line) the choice of a very coarse S h with a rather high p has been advocated [1], [2]. The use of wavelets on Σ has also been proposed and analyzed in [12].…”

“…One can also obviously lower the degree of the local polynomials instead of coarsening the mesh. However, the use of a high degree p (for the elements of Φ h ) together with a very coarse mesh was advocated for instance in [1], [2]. On the other hand, Assumption 1 would require, roughly speaking, that the degrees of freedom for V [2].…”

“…However, the use of a high degree p (for the elements of Φ h ) together with a very coarse mesh was advocated for instance in [1], [2]. On the other hand, Assumption 1 would require, roughly speaking, that the degrees of freedom for V [2]. Often it would be sufficient to lower the degree only for the elements near ∂Σ as it is done (roughly speaking) in the mortar method (see [8], [9], [33], [39]).…”

“…This amounts to add to E tot an interface contribution L = L(u, λ) := Σ λ · (u 1 − u 2 ) dS (1. 2) and to require the stationarity of the functional…”