Poincaé type inequalities for vector functions with zero mean normal traces on the boundary and applications to interpolation methods
Provides practical, explicit constants for finite element analysis, enabling sharper error bounds in numerical methods.
The paper derives explicit, computable bounds for Poincaré–Steklov constants for vector functions with zero mean normal traces on Lipschitz domains, including standard finite element shapes. These bounds are then applied to improve interpolation error estimates for scalar and vector functions.
In the paper, we consider inequalities of the Poincaré--Steklov type for subspaces of $H^1$-functions defined in a bounded domain $Ω\in \Rd$ with Lipschitz boundary $\partialΩ$. For scalar valued functions, the subspaces are defined by zero mean condition on $\partialΩ$ or on a part of $\partialΩ$ having positive $d-1$ measure. For vector valued functions, zero mean conditions are imposed on components (e.g., normal components) of the function on certain $d-1$ dimensional manifolds (e.g., on plane or curvilinear faces of $\partialΩ$). We find explicit and simply computable bounds of the respective constants for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions. %383838