arxiv: 2604.25797 · v1 · submitted 2026-04-28 · 💻 cs.CE

Recognition: unknown

Unfitted Multi-Level hp Refinement for Localized and Moving Solution Features

Jan Niklas Schm\"ake , Martin Ruess

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:51 UTC · model grok-4.3

classification 💻 cs.CE

keywords unfitted finite elementsmulti-level hp-refinementadaptive mesh refinementhp-adaptivitymoving sourceslocalized featuresfinite element analysisoverlay meshes

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The pith

Unfitted multi-level hp-refinement with independent overlay meshes captures localized and moving features efficiently without base mesh modifications.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops an adaptive finite element technique that adds refinement through overlay meshes positioned independently of the base mesh. The goal is to handle problems with sharp local changes or features that move over time without the overhead of repeatedly altering the main mesh or handling complex interfaces. By superimposing the approximation spaces and using constraints to maintain continuity, along with special integration over overlapping regions, the method couples the meshes accurately. Tests on benchmarks with discontinuities, singularities, and moving sources confirm that exponential convergence is preserved and that accuracy is achieved at lower computational cost than with aligned refinement approaches.

Core claim

The authors establish that a global C0-continuous approximation space can be constructed by superposing the active finite element spaces from a fixed base mesh and independently placed overlay meshes at multiple hp-refinement levels. Homogeneous constraints on the artificial boundaries of the overlays enforce continuity, while the system is assembled using integration domains defined by intersections of the element partitions. This unfitted construction permits refinement zones to be inserted, translated, and removed without altering the base discretization or requiring alignment of overlay boundaries with base mesh interfaces.

What carries the argument

The unfitted multi-level hp-refinement framework that superposes active spaces across refinement levels and assembles couplings over admissible intersection regions with homogeneous constraints for continuity.

If this is right

The unfitted approach retains exponential convergence for non-smooth problems such as those with discontinuities and singularities.
It achieves improved error-to-cost ratios compared to fitted multi-level hp-refinement strategies.
Comparable accuracy can be obtained with substantially fewer degrees of freedom in representative cases.
Localized high-order refinement can accurately track moving solution features over time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework may reduce the complexity of implementing adaptive simulations for problems with evolving domains or interfaces.
Extensions could include integration with other discretization techniques to handle even more complex moving features without custom remeshing algorithms.
Potential applications arise in engineering simulations where computational resources are limited but high local accuracy is needed around critical points.

Load-bearing premise

Homogeneous constraints on artificial overlay boundaries ensure global C0 continuity and that coupling assembled over intersections of element partitions produces accurate results without requiring alignment of overlay boundaries with the base mesh.

What would settle it

Demonstrating that the global solution loses C0 continuity or that accuracy degrades significantly for a non-smooth benchmark when overlay meshes are placed without alignment to the base mesh would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.25797 by Jan Niklas Schm\"ake, Martin Ruess.

**Figure 1.** Figure 1: Concept of refinement-by-superposition in one dimension. A locally refined overlay discretization is superimposed on a coarser base discretization, and the final coupled approximation 𝑢 is represented as the sum of the base contribution 𝑢𝑏 and the overlay contribution 𝑢𝑜 . Filled nodes denote active degrees of freedom, while open nodes are constrained to zero such that the overlay contribution vanishes at … view at source ↗

**Figure 2.** Figure 2: Schematic illustration of the fitted [29, 10, 30, 31, 32] multi-level hp-refinement in one dimension (a) and two dimensions (b). Global continuity and linear independence are enforced by deactivating selected shape functions, shown in gray in (a) and as gray surfaces, dotted lines, and missing vertices in (b). subsequent refinement levels, e.g., 𝑘 = 2. The bottom layer of view at source ↗

**Figure 3.** Figure 3: Schematics of the unfitted multi-level hp refinement in one dimension (a) and two dimensions (b). Blue and solid black components are active, while gray and dotted ones are deactivated. The vertical red surface indicates a line within the domain requiring increased resolution. three equally sized elements that fully cover the refinement region without aligning with base-mesh vertices. A second refinement l… view at source ↗

**Figure 4.** Figure 4: Visualization of the quadrature regions used for the evaluation of coupling terms between two finite element meshes. Each individual quadrature region is shown in a distinct color. Since this work focuses exclusively on axis-aligned meshes, the construction of integration regions can be realized by a simple dimension-independent algorithm, illustrated in view at source ↗

**Figure 5.** Figure 5: Visualization of the dimension-independent algorithm used to determine the intersection regions of two axis-aligned finite element meshes. The example shows two two-dimensional meshes (green and blue) on the left and the resulting 𝐶 ∞ integration regions on the right. Regions highlighted in red contain elements from both meshes. The example considers two partially overlapping meshes with non-matching eleme… view at source ↗

**Figure 6.** Figure 6: The one-dimensional bar problem. Shown in (a) is the geometric setup including boundary conditions; (b) displays the corresponding fitted multi-level hp mesh; and (c) shows the unfitted version. The bar is clamped at the left end and traction-free at the right. In its original form, the problem was subjected only to a sinusoidal volumetric force, resulting in a solution that cannot be represented exactly b… view at source ↗

**Figure 7.** Figure 7: Numerical strain solutions of the one-dimensional bar problem for four different meshes. Panels (a) and (b) correspond to single high-order elements with degrees 𝑝 = 10 and 𝑝 = 30, respectively. Panels (c) and (d) use local refinements: fitted multi-level hp in (c) and unfitted multi-level hp in (d). In all four plots, the exact solution from Equation (10) is shown in black and the numerical approximation … view at source ↗

**Figure 8.** Figure 8: Convergence of the relative energy-norm error ‖𝑒‖𝑟 for different refinement strategies applied to the one-dimensional bar problem. solutions. Since the unfitted approach permits explicit control over the size of the overlay meshes, three separate studies were performed using 𝛼 = {1∕3, 1∕2, 2∕3}, where 𝛼 = 2∕3 represents the largest permissible value for the given geometric configuration. The results for 𝛼 … view at source ↗

**Figure 9.** Figure 9: Geometric setup of the singular corner problem [32] in (a). (b) and (c) show the fitted and unfitted multilevel hp meshing approaches, respectively, each with three refinement levels toward the singularity. In the unfitted approach the overlay size is controlled by the parameter 𝛼. With the radial coordinate 𝜌 = ‖𝒙‖2 , the manufactured exact solution produces a singular gradient at 𝒙 = [0, 0]: 𝑢exact = √ … view at source ↗

**Figure 10.** Figure 10: Numerical solution of the singular corner problem for three different meshes. The top row displays the solution field 𝑢, while the bottom row depicts the gradient magnitude ‖∇𝑢‖2 . In all three solutions, a 1 × 1 base mesh and three subsequent refinement layers were used. The first column used fitted multi-level hp refinements, and the second and third used unfitted refinements with overlay size factors o… view at source ↗

**Figure 11.** Figure 11: Error convergence of the singular corner problem for different refinement strategies. Further reduction of the overlay size to 𝛼 = 0.2 improves the results, resulting in a significantly faster convergence rate and a final error of 0.12734 % with 1160 unknowns. These computations were performed without over-integration of the singular source term. While over-integration reduces the absolute errors, additio… view at source ↗

**Figure 12.** Figure 12: Setup and results of a test model used to investigate the influence of small overlaps between superimposed meshes. (a) shows the geometric setup, consisting of a 3 × 3 base mesh and a 2 × 2 overlay mesh offset in 𝑥1 and 𝑥2 by the scalar 𝜂. (b) displays the condition number 𝜅 of the resulting system matrix for different values of 𝑝 and 𝜂, and (c) shows the corresponding number of PCG iterations required t… view at source ↗

**Figure 13.** Figure 13: Geometric setup of the traveling heat source problem. Left: global computational domain. Right: magnified view of the traveling circular heat source. Inside this geometry, a small circular heat source moves along a circular path, thereby heating the domain. The governing initial-boundary-value problem is defined by 𝜕𝑇 𝜕𝑡 = ∇ ⋅ (𝜅 ∇𝑇 ) + 𝑠 ∀ ( 𝒙 ∈ Ω, 𝑡 ∈ (0, 𝑡max] ) , (15a) 𝑇 = 0 ∀ ( 𝒙 ∈ Γ𝐷, 𝑡 ∈ (0, 𝑡max] … view at source ↗

**Figure 14.** Figure 14: Numerical results for different meshing strategies of the traveling heat source problem at 𝑡 = 2.134. The top row shows the temperature field 𝑇 , the second row the temperature gradient magnitude ‖∇𝑇 ‖2 , and the bottom row the corresponding finite element meshes. The first column shows the unrefined 11 × 11 mesh with degree 𝑝 = 4, the second column the same base mesh with three unfitted overlay meshes sh… view at source ↗

**Figure 15.** Figure 15: Temperature 𝑇 (a) and temperature gradient magnitude ‖∇𝑇 ‖2 (b) probed at 𝒙 = (0, 2.5) for all computed time steps. The black curve shows the low-order reference solution, blue the unrefined high-order solution, and red the locally refined high-order solution. Panel (a) shows the temperature values over time: the reference solution in black, the unrefined high-order solution in blue, and the locally refin… view at source ↗

read the original abstract

Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through mesh-topology modifications, constraint handling for non-matching interfaces, and repeated remeshing with state transfer. This work presents an unfitted multi-level hp-refinement strategy that enriches a fixed base discretization by independently positioned overlay meshes. The global approximation space is constructed by superposition of the active spaces across all refinement levels, while homogeneous constraints on artificial overlay boundaries ensure global $C^0$ continuity. Coupling between non-matching meshes is assembled over admissible integration regions defined by intersections of element partitions, enabling reuse of standard element-level finite element routines within a lightweight superposition framework. In contrast to fitted multi-level approaches, overlay boundaries are not required to align with underlying mesh interfaces. This reduces inter-level coupling and allows refinement zones to be inserted, translated, and removed without modifying the base discretization. Numerical studies for discontinuous and singular benchmark problems, as well as a moving source, demonstrate the performance of the method. The unfitted approach retains exponential convergence for non-smooth problems and achieves improved error-to-cost ratios compared to fitted multi-level hp-refinement. For representative cases, comparable accuracy is obtained with substantially fewer degrees of freedom, while localized high-order refinement accurately tracks moving features.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The unfitted multi-level hp-refinement with non-aligning overlays is a practical extension that simplifies handling of moving localized features, and the reported numerics look consistent with the claims.

read the letter

The main new thing here is an unfitted version of multi-level hp-refinement. Overlay meshes sit independently on a fixed base discretization, the global space is built by superposition of active spaces from each level, and homogeneous constraints on the artificial overlay boundaries plus integration only over element-partition intersections are used to keep C0 continuity. This removes the need for overlay boundaries to line up with base-mesh interfaces, so refinement zones can be inserted, translated, or removed without touching the base mesh topology or doing state transfer. That is a clear step beyond the usual fitted multi-level hp setups that require alignment and more involved constraint handling. The numerical studies on discontinuous and singular benchmarks plus a moving source show exponential convergence is retained for non-smooth cases and that comparable accuracy is reached with substantially fewer degrees of freedom than the fitted counterpart. Those results are the strongest part of the abstract and make the efficiency claim concrete. The soft spot is exactly the one the stress-test flags: whether the homogeneous constraints actually produce a conforming space across non-matching hp-spaces of different degrees and whether the intersection-based quadrature preserves the expected rates without hidden errors. The abstract states that the studies confirm this, but the load-bearing step is the implementation detail on how the constraints couple the spaces and how the admissible regions are integrated. If those pieces hold up in the full text, the method works as described; if not, the convergence claims would need revisiting. This is aimed at people working on adaptive finite-element methods in computational mechanics who deal with localized or moving features. It is worth sending to peer review because the core construction is clearly motivated, the performance claims are specific and testable, and the approach is grounded in standard superposition and integration practices rather than new unverified machinery.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces an unfitted multi-level hp-refinement strategy for adaptive finite element discretizations of problems featuring localized singularities, discontinuities, sharp gradients, and moving sources. A fixed base discretization is enriched by independently positioned overlay meshes; the global approximation space is formed by superposition of active spaces from all levels, with homogeneous constraints imposed on artificial overlay boundaries to enforce global C0 continuity. Coupling is performed by assembly over admissible integration regions obtained from intersections of element partitions, allowing standard element routines to be reused without requiring overlay boundaries to align with the base mesh. Numerical studies on discontinuous and singular benchmark problems as well as a moving-source example are reported to demonstrate that the method retains exponential convergence for non-smooth solutions and yields improved error-to-cost ratios relative to fitted multi-level hp-refinement, achieving comparable accuracy with substantially fewer degrees of freedom.

Significance. If the central claims are substantiated, the approach would provide a practical simplification for adaptive hp-FEM by removing the need for repeated remeshing, topology modifications, and inter-level alignment, while preserving the exponential convergence properties that make hp-refinement attractive for non-smooth problems. The reuse of standard finite-element kernels inside a lightweight superposition framework and the ability to insert, translate, or remove refinement zones without altering the base mesh are concrete engineering advantages. The reported numerical comparisons to fitted counterparts, if quantified, would strengthen the case for the method in computational mechanics applications involving moving features or localized non-smoothness.

major comments (2)

[Abstract] Abstract: the central claims of retained exponential convergence for non-smooth problems and improved error-to-cost ratios rest on the numerical studies, yet the abstract supplies no quantitative data (convergence rates, error norms, DOF counts, or direct comparisons), rendering the magnitude of the reported gains unverifiable from the given information.
[Method description] Method description (space construction): the homogeneous constraints on artificial overlay boundaries are asserted to enforce global C0 continuity while coupling non-matching hp-spaces over intersection-based integration regions; because this step is load-bearing for conformity and for preservation of hp-approximation power, the manuscript must demonstrate (via explicit construction or numerical verification) that the constraints fully couple spaces of differing polynomial degrees without degrading the exponential rates or introducing quadrature artifacts.

minor comments (1)

[Abstract] Abstract: the phrase 'admissible integration regions defined by intersections of element partitions' would benefit from a short clarifying sentence or reference to the precise quadrature rule employed, to aid readers unfamiliar with unfitted intersection-based assembly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The two major comments identify areas where the manuscript can be strengthened for clarity and rigor. We address each point below and indicate planned revisions.

read point-by-point responses

Referee: [Abstract] Abstract: the central claims of retained exponential convergence for non-smooth problems and improved error-to-cost ratios rest on the numerical studies, yet the abstract supplies no quantitative data (convergence rates, error norms, DOF counts, or direct comparisons), rendering the magnitude of the reported gains unverifiable from the given information.

Authors: We agree that the abstract would benefit from concise quantitative indicators to allow readers to gauge the scale of the improvements directly. In the revised version we will incorporate selected quantitative highlights drawn from the numerical studies (e.g., observed exponential convergence behavior and the reported DOF reductions for comparable accuracy) while remaining within standard abstract length constraints. revision: yes
Referee: [Method description] Method description (space construction): the homogeneous constraints on artificial overlay boundaries are asserted to enforce global C0 continuity while coupling non-matching hp-spaces over intersection-based integration regions; because this step is load-bearing for conformity and for preservation of hp-approximation power, the manuscript must demonstrate (via explicit construction or numerical verification) that the constraints fully couple spaces of differing polynomial degrees without degrading the exponential rates or introducing quadrature artifacts.

Authors: The space construction, constraint imposition, and intersection-based assembly are presented in Sections 3–4. Global C0 continuity follows from the homogeneous Dirichlet constraints on artificial boundaries, and the formulation permits arbitrary polynomial degrees on overlay elements. Preservation of exponential hp-convergence is shown by the benchmark results in Section 5, which include cases with varying polynomial degrees across levels and exhibit the expected rates without visible degradation or quadrature-induced oscillations. To make the coupling property more explicit, we will add a short, isolated verification example (a simple patch test with deliberately mismatched degrees) that compares the constrained solution against a reference and confirms absence of artifacts. revision: partial

Circularity Check

0 steps flagged

No circularity: method construction and claims are independent of inputs

full rationale

The paper constructs an unfitted multi-level hp space via superposition of active overlay spaces with homogeneous boundary constraints and intersection-based assembly. These steps are presented as direct extensions of standard finite-element superposition and quadrature practices, without any equation reducing a claimed result (e.g., exponential convergence or error-to-cost improvement) to a fitted parameter or prior self-citation by construction. Numerical benchmarks on discontinuous, singular, and moving-source problems are used to demonstrate retention of hp-rates and DOF savings; no load-bearing premise is justified solely by self-reference or by renaming an input as a prediction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard finite element assumptions for function spaces, continuity, and numerical integration; no free parameters, new entities, or ad-hoc axioms are explicitly introduced in the abstract.

axioms (1)

standard math Standard assumptions of the finite element method hold, including suitable Sobolev spaces for the approximation and accurate quadrature over element intersections.
Implicit foundation for any FEM discretization paper; invoked when describing superposition and assembly.

pith-pipeline@v0.9.0 · 5533 in / 1337 out tokens · 55796 ms · 2026-05-07T13:51:50.841573+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

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Frits de Prenter, Clemens V. Verhoosel, Gert J. van Zwieten, and E. Harald van Brummelen. Condition number analysis and preconditioning of the finite cell method.Computer Methods in Applied Mechanics and Engineering, 316:297–327, 2017

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Frits de Prenter, Clemens V. Verhoosel, E. Harald van Brummelen, Mats G. Larson, and Santiago Badia. Stability and conditioning of immersed finite element methods: analysis and remedies.Archives of Computational Methods in Engineering, 30(6):3617–3656, 2023

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Stein K. F. Stoter, Sai C. Divi, E. Harald van Brummelen, Mats G. Larson, Frits de Prenter, and Clemens V. Verhoosel. Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics.Computer Methods in Applied Mechanics and Engineering, 412:116074, 2023

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Storti, Alejandro E

Bruno A. Storti, Alejandro E. Albanesi, Ignacio Peralta, Mario A. Storti, and Víctor D. Fachinotti. On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems.Finite Elements in Analysis and Design, 208:103789, 2022. 29 of 29

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