Boundary-preserving hp interpolation and p-robust discrete harmonic extensions on tetrahedral meshes

Abstract

We construct a boundary-preserving hp interpolation operator on three-dimensional tetrahedral meshes with locally variable polynomial degrees. If the trace of an H^1 function on the prescribed Dirichlet boundary is already a piecewise polynomial trace of the finite element space, the interpolant preserves this trace exactly and satisfies the standard local h_K/p_K approximation estimates. The statement follows the scaling form of Melenk's hp quasi-interpolation for nonsmooth functions. As a consequence, a discrete trace is extended by first applying the continuous trace theorem and then applying the boundary-preserving interpolant; the corresponding discrete harmonic extension is bounded by variational comparison. The proof of the interpolation theorem uses local polynomial trace liftings on tetrahedral boundary layers, nonsingular vertex patches, and a variable-degree tetrahedral projection. These auxiliary liftings are also stable in scaled boundary-layer norms.