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A unified approach for embedded boundary conditions for fourth-order elliptic problems. (English) Zbl 1352.65507
Summary: An efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier-Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche’s approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non-conforming meshes with non-interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B-splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J40 Boundary value problems for higher-order elliptic equations
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