Situan Li

Situan Li, Weiying Zheng

We establish weighted hp-uniform vertex-patch decompositions in arbitrary space dimension d >= 1 for tensor-product discretizations of H^k-type conforming and nonconforming spaces, with arbitrary fixed Sobolev order k >= 1, on fitted interface meshes. The cells are coordinate-compatible cuboids, the local spaces are Q_{p_K}(K) with arbitrary elementwise degrees satisfying p_K >= 2k-1, and the coefficient may have arbitrarily large jumps across material interfaces. Under local coefficient oscillation bounds and a local high-side connectivity condition, both the conforming H^k space and the nonconforming spaces V_h^{(s)}, 0 <= s <= k, admit stable decompositions with constants which may depend on the fixed parameters d and k, but are independent of the mesh size, all polynomial degrees, neighboring degree ratios, and the global coefficient contrast. The argument combines a Hermite endpoint transform for endpoint jets of order 0,...,k-1, its tensor-product extension, weighted broken patch Poincare inequalities, and a successive correction of normal derivative jumps. Numerical experiments for a three-dimensional DG problem with large coefficient jumps and strongly varying local polynomial degrees support the predicted robustness. For k = 1 the same conclusions hold on uniformly regular mapped cubical meshes whose neighboring element maps agree on each common face.

Situan Li, Weiying Zheng

We construct $h$- and $p$-robust, degree-preserving space decompositions and additive Schwarz preconditioners for variable-degree $hp$ finite element discretizations of reaction-diffusion and fitted-interface problems. On conforming simplicial meshes in arbitrary dimension, the single-domain result allows an arbitrary elementwise degree distribution subject only to $p_K\ge1$. A minimal-average Falk--Winther bubble transform is introduced by taking each subsimplex average over a fixed adjacent element of minimal polynomial degree. The resulting components remain in the prescribed variable-degree space and satisfy $L^2$- and $H^1$-stable estimates with constants independent of the mesh size, the polynomial degrees, and the way the degrees vary from element to element. Together with a stable continuous piecewise affine component, this yields an $hp$-uniform Schwarz preconditioner for single-domain reaction-diffusion problems with locally comparable coefficients. For three-dimensional fitted-interface problems, we use a symmetric Nitsche discretization on a tetrahedral mesh fitted to a piecewise planar interface. Surface jump components are lifted into the side selected by the penalty scaling, and the conforming remainder is decomposed by a weighted one-sided bubble transform. Grouping the components by vertices gives a practical vertex-patch Schwarz preconditioner. Under a common-degree condition on interface-touching tetrahedra, the condition number is bounded independently of the mesh size, the local polynomial degrees, the diffusion contrast, and the coefficient magnitudes. Numerical experiments for pure diffusion problems support the theory and suggest robustness beyond the common-degree assumption.

Situan Li, Weiying Zheng

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.

Situan Li, Weiying Zheng

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.