p-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving hp Interpolation on Tetrahedral Meshes

We construct p-robust polynomial trace liftings on three-dimensional tetrahedral meshes. The prescribed trace is a continuous piecewise polynomial function on a boundary face patch; the tetrahedra touching this patch have one common degree, while the interior degrees may be arbitrary. The lifting is degree-preserving, supported in the corresponding boundary layer, and satisfies both an H^1 estimate and a scaled boundary-layer L^2 estimate with constants independent of the mesh size and the polynomial degree. The construction is local and combines tetrahedral polynomial liftings, face-gluing arguments, and nonsingular vertex patches. As consequences of the construction, we obtain p-robust discrete harmonic extensions, including an H^1-seminorm-stable extension for the pure diffusion energy, and a boundary-preserving hp interpolation operator that keeps piecewise polynomial Dirichlet data exactly while retaining standard local approximation estimates.