Interpolation of functions based on Poincare type inequalities for functions with zero mean boundary traces

Abstract

We consider interpolation operators for scalar and vector valued functions based on mean values (or on mean values of normal components) defined on certain amount of faces of a mesh consisting of the Lipschitz subdomains (cells). The main result establishes sufficient conditions, which guarantee that a function u ∈ H¹ (vector function) can be interpolated in L² by a piecewise constant function (vector function) with minimal amount of parameters (degrees of freedom). It is proved that the difference between u and its interpolant is controlled by the norm of ∇u with a constant, which depends on the maximal diameter of cells forming the mesh. The method operates with minimal amount of interpolation parameters related to mean values on a certain amount of faces. For polygonal domains we deduce computable bounds of the interpolation constants, which are expressed throughout geometrical parameters of cells and show that they are close to sharp constants if they are known (see [8]). The interpolation method is not restricted to polygonal cells. We also present interpolation operators for cells with curvilinear boundaries and discuss possible extensions to meshes containing overlapping cells.