arXiv subjects

Situan Li

Publications and source records attributed to Situan Li.

6 recordsLinked to original sources

arXiv2026

hp-Optimal DG Approximation and Robust Schwarz Decompositions on One-Irregular Cubical Meshes

We study hp approximation and additive Schwarz decompositions for variable-order cubical finite element spaces on one-irregular meshes. For fitted homogeneous diffusion interface problems on one-irregular hexahedral meshes, we prove an hp-optimal energy-norm estimate for the interior penalty DG method. The interpolation input is a conforming hp interpolant obtained from fitted conforming closures of one-irregular vertex patches. We also derive stable decompositions for conforming and DG spaces. On one-irregular quadrilateral meshes the bounds allow locally comparable variable polynomial degrees and are independent of the mesh size, the local degrees, and, under a local coefficient quasi-monotonicity condition, the coefficient contrast. On one-irregular hexahedral meshes the conforming decomposition has the corresponding polylogarithmic loss; the DG-to-conforming reduction is used there for uniform-degree DG spaces. Numerical experiments illustrate the p-optimal DG error estimate and the robustness of the DG Schwarz preconditioner.

math.NA↗

arXiv2026

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.

math.NA↗

arXiv2026

Weighted hp-Uniform Decompositions for H^k-Type Tensor-Product Spaces in Arbitrary Dimension

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.

math.NA↗

arXiv2026

Owner-selected bubble transforms and coefficient-robust Schwarz preconditioners for variable-degree $hp$ finite elements

We construct $h$- and $p$-robust, degree-preserving space decompositions and additive Schwarz preconditioners for variable-degree $hp$ finite element discretizations of conforming reaction-diffusion and fitted-interface problems. On conforming simplicial meshes, an owner-selected Falk--Winther bubble transform gives $L^2$- and $H^1$-stable components with constants independent of the mesh size, the local polynomial degrees, and the degree distribution. Minimal-degree owners preserve arbitrary variable-degree spaces with $p_K\ge1$, while coefficient-adapted owners yield weighted estimates under local chain conditions. Combined with a weighted continuous piecewise affine extraction, this gives $hp$-uniform Schwarz preconditioners for conforming reaction-diffusion problems with locally comparable coefficients, and a coefficient-weighted conforming variant in the uniform-degree case. 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 using patch-level $p$-robust trace liftings. The conforming remainder is decomposed by the low-order extraction and a weighted one-sided bubble transform. Grouping the resulting components by vertices yields a practical vertex-patch Schwarz preconditioner whose condition number is independent of the mesh size, local polynomial degrees, diffusion contrast, and coefficient magnitudes under a common-degree condition on interface-touching tetrahedra. Numerical experiments for pure diffusion problems support the theory and suggest robustness beyond the common-degree assumption.

math.NA↗

arXiv2026

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

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.

math.NA↗

arXiv2026

Trace-Preserving hp Interpolation and Polynomial Liftings on Conforming Hexahedral Meshes

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.

math.NA↗