Stable Triangle Projections for Variable-Degree Tetrahedral Spaces and Uniform IPDG Preconditioning

The main ingredient of this paper is an edge-local variable-degree projection on a triangle that is uniformly stable in both L2 and H1. We use this two-dimensional operator in two tetrahedral constructions. First, on a reference tetrahedron, we build an H1-stable projection from a high order polynomial space onto a variable-degree space whose degrees are prescribed independently on edges, faces, and in the volume. Since the tetrahedral projection is local and trace-compatible, it also gives an h- and p-uniform stable decomposition, in the weighted energy norm, for conforming hp spaces, and hence a uniform additive Schwarz preconditioner for the conforming Laplace operator. Second, on a uniformly regular mapped tetrahedral mesh with elementwise variable polynomial degrees, the same triangular projection gives the finite-layer edge truncation needed in a p-uniform stable DG-to-CG decomposition for the symmetric IPDG norm. The DG-to-CG decomposition, combined with the conforming splitting, gives the IPDG preconditioner. The constants depend only on reference shapes, the local degree-spread bound within each tetrahedron, the neighbor-degree bound across mesh faces, uniform map-regularity, patch cardinalities, and the coefficient path constants; they are independent of h, of the local polynomial degrees, and of the coefficient contrast.