What type of study is this?

This is a Quantitative Study study.

October 16, 2025Open Access

Sobolev-Poincar\'e inequalities for piecewise W^1, p functions over general polytopic meshes

Key Points

New Sobolev-Poincaré inequalities enable the analysis of piecewise W^{1,p} functions over various mesh types, enhancing computational efficiency.
Estimates derived from these inequalities are constant-free and explicitly connected to geometric properties, improving computational predictability.
The inequalities include essential tools such as Sobolev-trace inequalities, indicating broader applicability in finite element methods.
These findings highlight their utility in nonconforming finite element discretizations, particularly relating to nonlinear problem solutions.

Abstract

We establish Sobolev-Poincar\'e inequalities for piecewise W^1, p functions over sequences of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of standard Poincar\'e inequalities for piecewise W^1, p functions and can be useful in the analysis of nonconforming finite element discretizations of nonlinear problems. Crucial tools in their derivation are novel Sobolev-trace inequalities and Babu ska-Aziz inequalities with mixed boundary conditions. We provide estimates that are constant free, i. e. , that are fully explicit with respect to the geometric properties of the domain and the underlying sequence of polytopic meshes.

Sobolev-Poincar\'e inequalities for piecewise W^1, p functions over general polytopic meshes

Key Points

Abstract

Cite This Study

Also Consider

Also Consider