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Abstract: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.

Submission history

From: Situan Li [view email]
[v1] Tue, 2 Jun 2026 04:31:15 UTC (200 KB)
[v2] Tue, 9 Jun 2026 14:08:32 UTC (207 KB)
[v3] Tue, 7 Jul 2026 11:07:02 UTC (1 KB) (withdrawn)