On the relaxation of continuity conditions for finite element schemes based on a least‐squares approach

The proposed contribution is based on the idea of the relaxation of continuity conditions and an enforcement of these continuity constraints for the considered fields via Lagrange multipliers. Therefore, a stress‐displacement least‐squares formulationℱ ( σ , u ) is considered, which is defined by the squared L 2 ( ℬ ) ‐norm applied to the first‐order system of differential equations, given by the balance of momentum and the constitutive equation, compare for the case of linear elasticity e.g. [1], as well as an additional (mathematically redundant) residual for enforcement of the stress symmetry condition. In general the assumptions for the continuity conditions are related to the conforming discretization of the individual fields. The conforming discretization, which demands continuity of the displacements and normal continuity of the stresses, is given by polynomial functions of Lagrange type for the displacements, i.e. u h ∈ P k , and a stress approximation with Raviart‐Thomas functions, i.e. σ h ∈ R T m . A non‐conforming discretization of the stresses considering discontinuous Raviart‐Thomas approximation functions with σ h ∈ ⅆ R T m yield to a relaxation of the continuity conditions. However the fulfillment of these relaxed constraints is enforced by the introduction of Lagrange multiplier within the underlying least‐squares formulation.