Poinca\'e type inequalities for vector functions with zero mean normal traces on the boundary and applications to interpolation methods

TL;DR

This paper establishes explicit Poincaré--Steklov inequalities for scalar and vector functions with zero mean boundary conditions, providing bounds useful in finite element methods and applications to function interpolation.

Contribution

It introduces new explicit bounds for Poincaré--Steklov constants for various domain types, including vector functions with boundary component conditions, and applies these to interpolation techniques.

Findings

Explicit bounds for Poincaré--Steklov constants on common finite element domains

Application of inequalities to improve interpolation methods

Extension of inequalities to vector functions with boundary component conditions

Abstract

In the paper, we consider inequalities of the Poincar\'e--Steklov type for subspaces of $H^1$-functions defined in a bounded domain $\Omega\in \Rd$ with Lipschitz boundary $\partial\Omega$. For scalar valued functions, the subspaces are defined by zero mean condition on $\partial\Omega$ or on a part of $\partial\Omega$ 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 $\partial\Omega$). 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