Mesh-Intrinsic GFEM: High-Order Smoothness on $C^0$ Unstructured Meshes
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914509245906944 |
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| author | Tian, Rong |
| author_facet | Tian, Rong |
| contents | High-order partial differential equations (PDEs) require derivative regularity that standard $C^0$ finite element infrastructures do not directly provide on unstructured meshes. We propose a mesh-intrinsic generalized finite element method (MiGFEM) that reconstructs local polynomial fields on overlapping nodal patches from shared nodal unknowns and blends them by a partition of unity, without introducing extra global degrees of freedom.
The core analysis establishes a partition-of-zero (PoZ) smoothness-transfer mechanism driven by interface coherence: derivative jumps cancel exactly for polynomial reproduction and decay as $O(h^{p+1-|α|}) $for smooth nonpolynomial fields. On this basis, we define a PoZ-consistent intrinsic derivative that is polynomial-exact and approximation-order consistent, enabling pointwise strong-form evaluation of high-order PDEs on $C^0$ meshes. For derivative-type/free boundary conditions in strong-form collocation,we introduce a boundary absorption constrained weighted least-squares strategy (BA-CWLS), which embeds boundary constraints into local patch reconstruction. This avoids globally overdetermined boundary augmentation and penalty tuning, while preserving a square sparse global system.
Numerical experiments show machine-precision patch tests,jump-decay rates consistent with theory, and robust performance on highly distorted meshes. The same mesh-intrinsic trial space supports both weak-form Galerkin and strong-form collocation discretizations, providing a unified high-order route on standard $C^0$ mesh infrastructures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23155 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mesh-Intrinsic GFEM: High-Order Smoothness on $C^0$ Unstructured Meshes Tian, Rong Numerical Analysis 65N30, 65N12, 65D05, 65N35 G.1.8; I.3.5 High-order partial differential equations (PDEs) require derivative regularity that standard $C^0$ finite element infrastructures do not directly provide on unstructured meshes. We propose a mesh-intrinsic generalized finite element method (MiGFEM) that reconstructs local polynomial fields on overlapping nodal patches from shared nodal unknowns and blends them by a partition of unity, without introducing extra global degrees of freedom. The core analysis establishes a partition-of-zero (PoZ) smoothness-transfer mechanism driven by interface coherence: derivative jumps cancel exactly for polynomial reproduction and decay as $O(h^{p+1-|α|}) $for smooth nonpolynomial fields. On this basis, we define a PoZ-consistent intrinsic derivative that is polynomial-exact and approximation-order consistent, enabling pointwise strong-form evaluation of high-order PDEs on $C^0$ meshes. For derivative-type/free boundary conditions in strong-form collocation,we introduce a boundary absorption constrained weighted least-squares strategy (BA-CWLS), which embeds boundary constraints into local patch reconstruction. This avoids globally overdetermined boundary augmentation and penalty tuning, while preserving a square sparse global system. Numerical experiments show machine-precision patch tests,jump-decay rates consistent with theory, and robust performance on highly distorted meshes. The same mesh-intrinsic trial space supports both weak-form Galerkin and strong-form collocation discretizations, providing a unified high-order route on standard $C^0$ mesh infrastructures. |
| title | Mesh-Intrinsic GFEM: High-Order Smoothness on $C^0$ Unstructured Meshes |
| topic | Numerical Analysis 65N30, 65N12, 65D05, 65N35 G.1.8; I.3.5 |
| url | https://arxiv.org/abs/2604.23155 |