Korn’s inequalities for piecewise $$H^1$$ vector fields.(English)Zbl 1055.65118

A generalized Korn’s inequality for piecewise linear vector fields with respect to simplicial triangulations is proposed. Here the analogy of the classical Korn’s inequalities for piecewise $$H^1$$ vector fields (functions) with respect to a partition of the domain consisting of now overlapping polyhedral subdomains is established for two- and three-dimensional domains, respectively.
Main result: A generalization of the author’s result with respect to a generalized Korn’s inequality for piecewise linear vector fields to piecewise polynomial vector fields which can be used to derive the classical Korn’s inequalities \begin{aligned} | u|_{H^1(\Omega,P)} &\leq C(\|\varepsilon_p(u)\|_{L^2(\Omega)}+ \| Qu\|_{L_2(\Omega)}),\\ | u|_{H^1(\Omega, P)} &\leq C\left(\|\varepsilon_p(u)\|_{L_2(\Omega)}+ \sup_{\substack{ m\in RM(\Omega),\\ \| m\|_{L_2(\Omega)= 1,\int_\Gamma mds= 0}}} \int u.mds\right)\end{aligned} $$(u\in [H^1(\Omega, P)]^d)$$, for some nonconforming finite elements that violate the weak continuity condition (mortar methods, discontinuous Galerkin methods).
Finally, the paper contains a discussion of the dependence of the constant in Korn’s inequality $$| u|_{H^1(\Omega)}\leq C_\Omega\|\varepsilon(u) \|_{L_2(\Omega)} \forall u\in [H^1(\Omega)]^d$$, $$|\int_\Omega \nabla xu\,dx|= 0$$, $$d= 2,3$$ on a two- or three-dimensional $$\Omega$$.

MSC:

 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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