Trace-Preserving hp Interpolation and Polynomial Liftings on Conforming Hexahedral Meshes
Provides a key local ingredient for high-order finite element analysis on conforming hexahedral meshes, enabling stable polynomial extensions and interpolation with trace preservation.
The paper constructs p-robust polynomial liftings on conforming hexahedral boundary patches that preserve prescribed edge and face traces, with stable H^1 norm and seminorm estimates. These liftings yield trace-preserving interpolation operators for H^1 functions, with p-uniform constants under uniform boundary-degree conditions and polylogarithmic loss otherwise.
Trace-compatible polynomial extensions are a recurring local ingredient in high-order finite element analysis on conforming hexahedral meshes. They are needed whenever prescribed edge and face traces must be preserved while a polynomial is extended into a neighboring cell or boundary patch. The main contribution of this paper is the construction of p-robust polynomial liftings on nonsingular conforming hexahedral boundary patches, with stable control of both the H^1 norm and the H^1-seminorm estimates needed for energy arguments. These liftings imply H^1-seminorm stable discrete harmonic extensions of polynomial Dirichlet traces. They also serve as boundary corrections for the conforming hp Clement interpolant, yielding trace-preserving interpolation operators for functions with only H^1 regularity. Under the uniform boundary-degree condition the constants are p-uniform; in the non-uniform case the stated logarithmic loss appears. We also treat meshes that may contain conforming singular boundary patches, where the loss remains polylogarithmic in the maximal local degree. Trace-preserving interpolation on reference cells and vertex-supported decompositions are developed as local tools for these patch and mesh-level constructions.