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UnCut FEM with PHT-splines on unfitted T-meshes

Falai Chen, Nuo Lei, Buyang Li

arXiv:2610.09909v1math.NAcs.NA

TL;DR

Body-fitted meshes and geometry-dependent integration pose challenges for PDEs on complex or evolving domains. The paper combines C1 PHT-splines with a least-squares UnCut method assembled on the background mesh. It proves well-posedness and optimal-order H1 estimates, and reports an effective residual-based criterion for adaptive refinement.

  • Problem

    Body-fitted discretizations can make high-quality meshes costly to construct, while unfitted methods may require difficult geometric integration over cut elements.

    Source: §1 · pp. 1–2

  • Method

    The method uses C1 PHT-splines to minimize the PDE’s L2 residual in a fully primal UnCut formulation assembled over the background mesh.

    Source: §6 · p. 21

  • Results

    The authors prove existence and uniqueness and optimal-order H1 error estimates, and report that the residual-based indicator is effective for adaptive refinement.

    Source: §1 · p. 3, §6 · p. 22

  • Takeaways & Limitations

    Standard element-wise quadrature on the active background domain avoids geometric integration over curved boundaries or cut subregions.

    Source: §6 · p. 21

    The main results assume polynomial degree k≥3 for global C1 continuity; the interpolation constant may depend on higher-order Sobolev norms of the level-set function.

    Source: §2 · p. 4, §3.1 · p. 9

Abstract

from arXiv · show

We propose a novel unfitted finite element method based on polynomial splines over hierarchical T-meshes (PHT-splines) for elliptic partial differential equations (PDEs) posed on domains whose boundaries are described implicitly by a level-set function. Exploiting the enhanced smoothness of the spline space, we formulate the method by minimizing the $L^2$ residual of the PDE over an extended neighborhood of the physical domain. This formulation allows the method to be implemented entirely on the background mesh using standard element-wise quadrature. It yields a symmetric positive-definite linear system and avoids the geometric construction of curved cut subregions in boundary elements, as well as the associated nontrivial numerical integration. We establish well-posedness of the numerical scheme and derive optimal \emph{a priori} error estimates in the $H^1$ norm. In addition, we employ a residual-based \emph{a posteriori} error indicator to guide local adaptive refinement of the hierarchical T-meshes. Numerical experiments corroborate the theoretical results and demonstrate the accuracy and efficiency of the proposed approach.

1. Introduction

Unfitted methods can avoid body-fitted mesh construction, but cut elements bring conditioning and geometric-integration challenges. This paper uses C1 PHT-splines in a least-squares UnCut formulation and establishes its well-posedness, optimal H1 error estimates, and residual-guided adaptivity.

1. Introduction

  • Body-fitted meshing can be costly for complex or evolving domains, while unfitted methods face small-cut-cell conditioning and difficult geometry-dependent quadrature.Repeated mesh generation can also eclipse solver costs in multi-query settings, and moving interfaces may require remeshing.Source: §1 · pp. 1–2
  • Earlier UnCut approaches address geometry-dependent integration, but parameter selection or an H−1-residual mixed formulation can add difficulty or computational cost.ϕ-FEM stability parameters may be challenging to choose for general elliptic problems; the parameter-free H−1 approach uses an auxiliary variable.Source: §1 · p. 2
  • The proposed method minimizes the PDE’s L2 residual with C1 PHT-splines, assembling over the background mesh with standard quadrature and no dual variables or cut-subregion integration.The formulation is fully primal and uses the spline space’s C1 continuity to discretize the PDE in strong form.Source: §1 · pp. 2–3
  • The authors prove existence and uniqueness of the discrete solution and derive optimal-order error estimates in the H1 norm.The analysis is needed because minimizing an L2 residual does not directly provide an H1-coercive variational structure.Source: §1 · p. 3
  • Hierarchical T-meshes support local refinement, and the proposed residual-based a posteriori indicator is demonstrated as an effective refinement criterion.The mesh permits T-junctions and uses cross-insertion for local h-refinement.Source: §1 · pp. 3–4

2. UnCut FEM for the Poisson equation

The section defines PHT-spline spaces on hierarchical T-meshes and constructs an unfitted Poisson scheme using an approximate level-set interface. Under stated assumptions, the scheme has a unique solution and an H1-error estimate.

2. UnCut FEM for the Poisson equation

  • For k≥3, the PHT-spline space is globally C1-continuous and supports local refinement while preserving linear independence and interface coupling.T-junctions add no independent degrees of freedom; Bézier extraction enforces C1 constraints across refinement levels.Source: §2 · p. 4

2.1. Unfitted mesh.

  • The discrete interface Γh is the zero set of the level-set approximation, and cut cells are identified within the retained background-mesh submesh.The extended domain is the union of supports of basis functions intersecting Ωh.Source: §2.1 · pp. 4–5

2.2. A simple least-squares UnCut FEM using PHT-splines.

  • Because Sh is globally C1-continuous, the scheme's jump penalty involves only higher-order derivatives, and computations use standard element-wise quadrature on the background mesh.This avoids numerical integration over geometrically curved cut elements.Source: §2.2 · p. 6
  • Under Assumption 2.1 and Ω⊂Ωh, the scheme has a unique solution and satisfies an H1-error estimate.The a priori analysis additionally assumes globally comparable active meshes; the locally refined meshes studied for adaptation need not satisfy that condition.Source: §2.2 · pp. 6–7

3. Proof of Theorem 2.4

The proof develops stability and approximation tools to establish well-posedness and derive error estimates using energy, H2, and duality arguments. For polynomial degree k ≥ 2 on a sufficiently refined mesh, it proves an optimal global H1 error bound by controlling boundary and gradient errors and extending the estimate to the computational domain.

3.1. Technical results.

  • A stability property for functions in Sh and interpolation estimates provide technical ingredients for the subsequent well-posedness and error analysis.The section introduces the stability result and a corollary giving interpolation bounds in a broken Sobolev norm.Source: §3.1 · pp. 7, 11
  • The modified spline interpolant yields error estimates for m = 1, 2, with constants that may depend on higher-order Sobolev norms of the level-set function.The proof treats patches far from the interface separately from those in an O(h)-wide strip, then derives broken-norm interpolation estimates under the stated geometric and mesh assumptions.Source: §3.1 · pp. 9–11

3.2. Stability and well-posedness.

  • The bilinear form is symmetric positive definite on Sh, and continuity of the linear functional yields a unique discrete solution by the Riesz representation theorem.Source: §3.2 · p. 11

3.3. H1-error estimate.

  • The proof then sets up a dual problem and uses a duality argument to pursue the optimal H1-error bound.The supplied passages outline this final step but do not include its completed estimate.Source: §3.3 · pp. 12–13
  • The error proof first obtains an energy-norm estimate for the discrete error component and then derives the final optimal H2-norm error estimate.Source: §3.3 · p. 13
  • The boundary-error estimate absorbs higher-order terms for k ≥2, using the previously established H2-error bound and elliptic regularity.On Γ, eh = −uh = (ϕ − ϕh)wh, which provides the starting decomposition for the estimate.Source: §3.3 · pp. 14–15
  • Collecting the estimates gives a comprehensive gradient-error bound over physical domain Ω, with the final bound Ch^(k+1)||f||H^(k−1)(Ωh) + Ch||eh||H1(Ωh).The stabilization penalty for the dual interpolant combines low- and high-order jump contributions. High-order jumps 3 ≤ j ≤ 2k use discrete scaled trace and inverse inequalities; for the low-order jump j = 2, v ∈ H3(eΩh) ensures continuity of the exact normal derivative across faces.Source: §3.3 · pp. 15–17
  • The proof extends the physical-domain gradient bound to Ωh using the ghost-penalty discrete extension property and a generalized Poincaré–Friedrichs inequality.It also uses the interfacial L2 estimate and controls the interpolation trace error before absorbing the remaining global H1-error terms.Source: §3.3 · p. 17
  • For k ≥2 and sufficiently small h, the global error satisfies ∥eh∥H1(Ωh) ≤ Ch^k∥f∥H^(k−1)(Ωh).The proof absorbs remaining error terms using k − 3/2 > 0 and Young’s inequality.Source: §3.3 · p. 17

4. A posteriori error analysis

The a posteriori analysis establishes two-sided bounds for the Laplacian error using an interior L2-residual indicator, then uses it to guide local refinement of the hierarchical T-mesh.

4.1. Reliability and efficiency of the error indicator.

  • Global C1 continuity removes the need for interface jump residuals, allowing an a posteriori indicator based exclusively on the interior L2-residual.Source: §4.1 · p. 18
  • The indicator η bounds the Laplacian error from above and below on Ωh, up to the perturbation term ∥r∥L2(Ωh).The proof obtains the bounds by applying the triangle inequality to the residual and perturbation.Source: §4.1 · p. 18

4.2. Adaptive algorithm.

  • The adaptive loop repeats solve, estimate, mark, and refine until the error tolerance is met, using Dörfler marking to select cells contributing significantly to the global indicator.Source: §4.2 · pp. 18–19
  • Marked cells are subdivided one-to-four in 2D or one-to-eight in 3D, with neighboring cells recursively refined to enforce 1-regularity.Source: §4.2 · pp. 18–19

5. Numerical experiments

Numerical experiments show the proposed method's convergence on uniform and adaptive meshes in two and three dimensions. Adaptive refinement reduces the degrees of freedom needed for comparable accuracy while retaining the reported optimal rates.

5. Numerical experiments

  • Experiments use bicubic PHT-splines (k = 3), tolerance 10^-8, marking parameter 0.25, and at most 20 iterations unless stated otherwise.The implementation uses IGAPack and the hierarchical T-mesh.Source: §5 · p. 19

5.1. Example 1.

  • Uniform refinement on the circular level-set test (r = 0.3) achieves O(h3) H1 and O(h2) H2 convergence.Errors are evaluated on the background mesh using an exact solution defined over the entire background domain.Source: §5.1 · p. 19, Table 1 · p. 20

5.2. Example 2.

  • Nearly 60% fewer DOFs (6128 vs. 14764) yield comparable accuracy under adaptivity; at fixed DOFs, error falls almost an order of magnitude.The 3328-leaf-cell adaptive mesh concentrates refinement near the sharp peak at (0.3, 0.7).Source: Table 2 · p. 20, §5.2 · p. 20, Figure 2 · p. 20

5.3. Example 3.

  • In three dimensions, H1 and H2 errors retain optimal asymptotic convergence rates on both uniform and adaptive meshes.The reported rates persist under aggressive local refinement and in the presence of hanging nodes.Source: §5.3 · p. 21

6. Conclusion

The paper presents a least-squares UnCut FEM for level-set Poisson problems using C1-continuous PHT-splines. Its analysis establishes well-posedness and optimal-order H1 error estimates, while experiments support effective adaptive refinement.

6. Conclusion

  • The method minimizes the strong-form L2 residual using C1-continuous PHT-splines and assembles on the active background domain with standard element-wise quadrature.This avoids geometric integration over curved boundaries or cut subregions and yields a fully primal formulation.Source: §6 · p. 21
  • The analysis proves existence and uniqueness and establishes optimal-order a priori error estimates in the H1 norm.A residual-based a posteriori indicator guides local refinement and is reported to provide an effective criterion.Source: §6 · pp. 21–22
  • The proposed framework is planned for future extension to PDEs on surfaces and trimmed CAD geometries.The stated future direction leverages PHT-splines' exact geometric representation capabilities.Source: §6 · p. 22
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