Hierarchical Hybrid Grids Overview

• Hierarchical hybrid grids are mesh constructions that combine an unstructured coarse mesh with structured, uniform refinement to create a multilevel hierarchy ideal for efficient geometric multigrid solvers and adaptive algorithms.
• They leverage hierarchical basis functions, simple binary intergrid transfer operators, and patchwise additive Schwarz smoothing to achieve convergence rates that are independent of mesh size and, in many cases, the polynomial order.
• These grids underpin scalable frameworks such as HyTeG and support massive simulations—demonstrated on systems with up to 98,304 compute cores and billions of degrees of freedom—while also enabling adaptive extensions like kℓ-refinement and agglomeration.

Searching arXiv for recent and directly relevant papers on hierarchical hybrid grids. {"^^^^^^^^2query2 hybrid grids\"","max_results":^^^^2all:\2query2^^^^,"sort_by":"submittedDate","sort_order":"descending"} Here are the search results from arXiv:

^^^^2all:\2^^^^. "kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids" (&&&^^^^2query2^^^^&&&) Authors: Daniel Koestler, Florian Tobias Schuster, Christian Godenschwager, Ulrich Ruede Published: 2^^^^2query22^^^^5-^^^^2query2 URL: http://arxiv.org/abs/(&&&^^^^2query2^^^^&&&)v^^^^2all:\2^^^^ Categories: cs.MS, math.NA

1. "H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation" (Jiang et al., 2024) Authors: Shuhao Song, Xinyi Zheng, Qiyu Niu, Yi Xu, Juncai Liang, Binqiang Zhao, Neng Wang, Yilong Chen, Zhenzhong Chen Published: 2^^^^2query22^^^^4-^^^^2query2 URL: http://arxiv.org/abs/([2403.10821](/papers/2403.10821))v^^^^2all:\2^^^^ Categories: cs.RO, cs.CV
2. "Non-invasive multigrid for semi-structured grids" (Mayr et al., 2021) Authors: Edward A. Phipps, William Spotz, James Hu, Mark Hoemmen Published: 2^^^^2query22all:\2^^^^-^^^^2query2 URL: http://arxiv.org/abs/([2103.11962](/papers/2103.11962))v^^^^2all:\2^^^^ Categories: cs.MS, math.NA
3. "Computing homogenized coefficients via multiscale representation and hierarchical hybrid grids" (Hannukainen et al., 2019) Authors: Jean-Christophe Mourrat, Felix Otto, Timo Schweitzer Published: 2^^^^2query2all:\2^^^^9-^^^^2query2 URL: http://arxiv.org/abs/([1905.06751](/papers/1905.06751))v^^^^2all:\2^^^^ Categories: math.NA
4. "A Scalable and Modular Software Architecture for Finite Elements on Hierarchical Hybrid Grids" (Kohl et al., 2018) Authors: Christian Godenschwager, Mathias Bauer, Harald Köstler, Ulrich Rüde Published: 2^^^^2query2all:\2^^^^8-^^^^2query2 URL: http://arxiv.org/abs/([1805.10167](/papers/1805.10167))v^^^^2all:\2^^^^ Categories: cs.MS

Need maybe search broader to include finite cell method hierarchical multigrid maybe not exact phrase. Search "Hierarchical Hybrid Grids multigrid". Searching for multigrid and hierarchical/hybrid grid variants closely related to the topic. {"^^^^^^^^2query2 hybrid grids OR hybrid grids) AND multigrid","max_results":^^^^2all:\2query2^^^^,"sort_by":"submittedDate","sort_order":"descending"} Here are the search results from arXiv:

^^^^2all:\2^^^^. "Geometric Multigrid solvers for Hybrid High-Order methods on polytopal meshes" (&&&^^^^2all:\2query2^^^^&&&) Authors: Daniele A. Di Pietro, Jakub Droniou, Tadele Kena, Tso Nsing Published: 2^^^^2query22^^^^6-^^^^2query2 URL: http://arxiv.org/abs/(&&&^^^^2all:\2query2^^^^&&&)v^^^^2all:\2^^^^ Categories: math.NA, cs.MS

1. "kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids" (&&&^^^^2query2^^^^&&&) Authors: Daniel Koestler, Florian Tobias Schuster, Christian Godenschwager, Ulrich Ruede Published: 2^^^^2query22^^^^5-^^^^2query2 URL: http://arxiv.org/abs/(&&&^^^^2query2^^^^&&&)v^^^^2all:\2^^^^ Categories: cs.MS, math.NA
2. "Towards Robust Geometric Multigrid For Complicated FE Spaces" (&&&^^^^2all:\2^^^^4&&&) Authors: Linus Seelinger, M. M. Chinthalapati, Peter Vaněk Published: 2^^^^2query22^^^^3-^^^^2all:\22 URL: http://arxiv.org/abs/(&&&^^^^2all:\2^^^^4&&&)v^^^^2all:\2^^^^ Categories: math.NA
3. "Non-invasive multigrid for semi-structured grids" (Mayr et al., 2021) Authors: Edward A. Phipps, William Spotz, James Hu, Mark Hoemmen Published: 2^^^^2query22all:\2^^^^-^^^^2query2 URL: http://arxiv.org/abs/([2103.11962](/papers/2103.11962))v^^^^2all:\2^^^^ Categories: cs.MS, math.NA
4. "Hierarchical multigrid approaches for the finite cell method on uniform and multi-level hp-refined grids" (&&&^^^^2all:\2^^^^8&&&) Authors: Markus Ruess, Bernhard Zwicknagl, Timon Rabczuk, Alexander Düster Published: 2^^^^2query22query2^^^^-^^^^2all:\2query2 URL: http://arxiv.org/abs/(&&&^^^^2all:\2^^^^8&&&)v^^^^2all:\2^^^^ Categories: cs.MS, math.NA
5. "Second Order Multipoint Flux Mixed Hybrid Finite Element Methods for Hybrid Grids" (&&&2^^^^2query2^^^^&&&) Authors: Jan M. Nordbotten, Ivar Aavatsmark Published: 2^^^^2query2all:\2query2^^^^-^^^^2all:\2query2 URL: http://arxiv.org/abs/(&&&2^^^^2query2^^^^&&&)v^^^^2all:\2^^^^ Categories: math.NA

Need maybe search "hierarchical hybrid grids software architecture finite elements". Already exact search enough. We can answer grounded in provided data and maybe mention additional R3MG? Search "agglomeration hierarchical hybrid grids". But perhaps enough. Maybe search broader for radiative transfer hierarchical grids? But the task is encyclopedia article on Hierarchical Hybrid Grids; broader usages might benefit from search exact phrase plus others maybe not necessary. Since prompt includes a data block with many papers. Need write article only, not mention search. Citations in article should include arXiv ids.

Need think structure: Intro para define HHGs: combine unstructured coarse meshes with structured refinement; in PDE/HPC constructed by uniformly refining coarse unstructured mesh, enabling structured-grid kernels and multigrid while preserving geometric flexibility [(Kohl et al., 2018) maybe 2^^^^2all:\2query2^^^^3]. Also term used in finite-element method with locally uniform partitions and in related hierarchical immersed discretizations (Hannukainen et al., 2019, &&&^^^^2all:\2^^^^8&&&). Then numbered h2 sections:

^^^^2all:\2^^^^. Core definition and construction

facts from HHG papers: unstructured coarse, structured fine, macro-primitives etc.

2. Algebraic and geometric structure

include nested spaces, locally uniform partition, only reference simplex stored, DOF organization maybe from ^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5 and 2^^^^2query2all:\2query2^^^^ and 2^^^^2all:\2query2^^^^3.

3. Multigrid and matrix-free solvers

combine 2^^^^2query2all:\2query2^^^^, 2^^^^2all:\2query2^^^^3, ^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5, maybe ^^^^2all:\2sort_order2query2^^^^7 quasi-matrix-free hybrid multigrid on adaptive Cartesian grids though phrase hybrid multigrid rather than HHG. But as related usage of hybrid approach. Need be careful concrete claims are from data. Could say "In one line of work, multigrid solvers for hierarchical hybrid grids have been proposed to promote the efficient utilization of high performance computer architectures" directly from 2^^^^2all:\2query2^^^^3 abstract. Then detail elementwise/patchwise AS smoothing from 2^^^^2query2all:\2query2^^^^, region-based multigrid from 2^^^^2all:\2query2^^^^3, operator surplus from ^^^^2all:\2sort_order2query2^^^^7 as related but not HHG proper maybe "Related hybrid multigrid realizations on dynamically adaptive Cartesian grids store hierarchical surpluses ..." etc.

4. Software architecture and scalability

HyTeG macro-primitive graph, distributed data, load balancing, MPI/OpenMP, weak scaling, billion unknowns [^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5, 2^^^^2query2all:\2query2^^^^].

5. Adaptivity and generalizations

kℓ-refinement from 25^^^^2query2^^^^8, semi-structured grids from 2^^^^2all:\2query2^^^^3, polytopal/agglomerated hierarchies from 26^^^^2query2^^^^3 and R3MG 24^^^^2query2^^^^4 as related extensions. Need mark broadening beyond classical HHG perhaps "Related developments generalize the HHG idea..."

6. Applications and broader uses

homogenization [^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5], elasticity, Stokes, geophysics [^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5, ^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^8], real-time dense mapping H3 [24^^^^2query2^^^^3] perhaps note "Outside PDE solvers, the phrase hierarchical hybrid representation denotes ...". Could also mention spatial index (0705.0204)? But title not HHG exactly "Using Images to create a Hierarchical Grid Spatial Index". Maybe not. Yet user topic is broad phrase maybe include broader domains. But if including, need articulate carefully as "Outside numerical PDEs, hierarchical hybrid grid ideas also appear as fixed-grid plus hierarchical subdivision spatial indexing". Since the data block includes this. Could have a section on "Related but non-identical usages" maybe mention because term spans domains. Need objective note that terminology is not uniform.

Need address misconceptions/controversies. Could mention that HHGs are not fully unstructured meshes nor purely structured grids; that's supported by repeated descriptions. Another misconception: matrix-free does not imply absence of operator data; in quasi-matrix-free approach, "the grid carries the matrix" [^^^^2all:\2sort_order2query2^^^^7]. Another: standard smoothers suffice? But in finite cell problems standard smoothers ineffective due to cut cells [2^^^^2query2all:\2query2^^^^]. Need maybe table. Could have table of representative constructions across domains. 3 cols. Keep brief. Since article comprehensive; maybe a small table helpful. But detailed discussion outside.

Need cite each paper at most once per section. Introduction can cite maybe ^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5 and ^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5 only. Each section max one citation per paper. Fine.

Need avoid invented statistics beyond data. Data includes 98,3^^^^2query2^^^^4 compute cores, 3.2 billion DOFs, etc. Use exact phrases. Could mention "with 3.2 billion DOFs" from 2^^^^2query2all:\2query2^^^^ details. Need ensure if claim about "billions of DOFs and nearly ^^^^2all:\2query2query2^^^^,^^^^2query2query2query2^^^^ cores" exact. Could use those exact numbers. Similar ^^^^2all:\222^^^^88 cores etc from ^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^8.

Need "Every concrete claim must appear verbatim in data block." Means any workflow step, framework names, etc. Need stick closely. I can paraphrase but maybe safe to use direct phrasing. Yet "verbatim" appears tough—"must appear verbatim in data block." Maybe easier to use phrases and equations directly from provided details. Avoid new adjectives like "classical" if not in data? Actually appears in data. Avoid "canonical". Maybe okay if inferential marked. We need factual fidelity. So any claim should reflect data. Probably okay to paraphrase if same meaning? But says verbatim. Better to adhere closely to wording from supplied details. So article sentences should be assembled from data content. That's restrictive. Need ensure claims only from data. I'll avoid deriving novel comparisons.

Need mention mathematical notation from source exactly. E.g., PRESERVED_PLACEHOLDER_^^^^2query2^^^^, etc. Use formulas given. Need maybe define nested spaces and restriction/prolongation in section 2 or 3.

Let's draft carefully with direct wording: Intro: "Hierarchical hybrid grids (HHGs) combine an unstructured coarse mesh with structured uniform refinement within each element, producing a multilevel, hierarchical mesh. In finite-element and multigrid settings, this hybrid grid offers the geometric flexibility of unstructured meshes with the computational efficiency typical for structured grids, and it is used to support matrix-free multigrid implementations with excellent performance (Kohl et al., 2018). Related formulations use an unstructured coarse grid to represent the variations of the coefficient field and refine each coarse element in a 'fully structured' manner, allowing very significant gains in memory usage (Hannukainen et al., 2019)."

Need section numbering in h2 title like "## ^^^^2all:\2^^^^. Construction and hierarchy" etc.

Section^^^^2all:\2^^^^ can talk classical HHG construction:

• Begin coarse, possibly unstructured simplicial macro-triangulation PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^, uniform refinement multiple times, $h=2^{-l}H$, macro-primitives graph, locally uniform partition $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$, only reference simplex fine mesh explicitly stored, reconstructed on-the-fly [^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5 and ^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5]. Maybe cite both in section once each. Need maybe include small table comparing constructions: Classical HHG, Locally uniform partition, Semi-structured generalization. Since data in papers. But section details okay in paragraphs.

Section2 basis and discrete operators:

• For multilevel hp and finite cell, hierarchical basis with integrated Legendre polynomials $N_j(x)=\int_{-1}^x L_j(t)dt$; overlay meshes; nested spaces $V_0 \subset ...\subset V_\ell$. Restriction/prolongation binary selection matrices no explicit assembly. Matrix structure block form. Could mention these as "hierarchical multigrid on uniform and multi-level hp-refined grids", maybe article about HHGs more broadly, but title hierarchical hybrid grids may refer to multi-level hp ("hierarchical hybrid"). Need include because prompt topic singular broad. Cite 2^^^^2query2all:\2query2^^^^. Also maybe quasi-matrix-free hybrid multigrid on dynamically adaptive Cartesian grids: spacetree-based multiscale cascade, hierarchical surpluses $\hat A = A - A_{rediscretized}$, etc. This is more "hybrid multigrid" than hybrid grids. Yet relevant to operator storage across hierarchical grids. Could include in section 3 as related operator-centric construction.

Section3 multigrid, smoothers, semi-structured generalization.

• Classical V-cycle $n_s$ pre/post smoothing.
• Standard smoothers ineffective for FCM due to cut cells; additive Schwarz formula; elementwise uniform, patchwise multi-level hp. Convergence independent of mesh size $h$, cut configuration, and patchwise AS independent of $p$ in 2D and close to independent in 3D. [2^^^^2query2all:\2query2^^^^]
• Non-invasive multigrid generalizes HHG to semi-structured grids; domain decomposition PRESERVED_PLACEHOLDER_^^^^2all:\2query2^^^^; PRESERVED_PLACEHOLDER_^^^^2all:\2all:\2^^^^, regional matrices with PRESERVED_PLACEHOLDER_^^^^2all:\22^^^^; exact equivalence to classical multigrid when interface stencils and aggregation match [2^^^^2all:\2query2^^^^3]. Maybe section title includes "Multigrid formulations and smoothers." Could cite both once. Need maybe mention related spacetree-based multigrid and operator surplus as adjacent development [^^^^2all:\2sort_order2query2^^^^7]. But if too much may distract.

Section4 software architecture and scalability

• HyTeG new generic higher-order finite-element framework for massively parallel simulations [^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5]. macro-primitives, distributed data, ghost layers, control/packing/buffer layers, load balancing partitioners ParMETIS, Scotch, Zoltan. matrix-free operator eval and multigrid smoothers using stencils.
• finite cell implementation with MPI, OpenMP, Trilinos (Epetra, AztecOO); AS smoothing parallelizable; popcorn benchmark up to 98,3^^^^2query2^^^^4 compute cores with 3.2 billion DOFs [2^^^^2query2all:\2query2^^^^].
• stencil scaling method on HHGs variable coefficient vector-valued PDEs, max speedups of 64% and ^^^^2all:\222^^^^% compared to on-the-fly integration; largest example ^^^^2all:\222^^^^88 compute cores on SuperMUC-NG [^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^8]. Need exactly. Could include. Need be careful "maximum speedups of 64% and ^^^^2all:\222^^^^%" from abstract. Good.

Section5 adaptivity and generalizations

• kℓ-refinement introduced adaptive mesh refinement for HHG: classical unstructured refinement only on coarsest level, fixed number of regular structured refinements all macro elements, cheap error estimator from full multigrid approximations, implemented in HyTeG, weak scaling up to billions DoFs, time spent in AMR and error estimation sub-^^^^2all:\2^^^^% [25^^^^2query2^^^^8]. Need claim "sub-^^^^2all:\2^^^^%" from details yes. Good.
• semi-structured grids as broader community and non-invasive adoption [2^^^^2all:\2query2^^^^3] maybe already used in section 3; can mention here maybe not repeat. Use 26^^^^2query2^^^^3 too: first optimal geometric multigrid for HHO discretizations on arbitrary polytopal agglomeration hierarchies in 2D and 3D with modified skeleton spaces. This isn't HHG proper but generalization of hierarchical hybrid/hybrid discretization. Need mark as plausible implication or related extension. Since title "hybrid high-order methods on polytopal meshes"; may still relevant due "hybrid" but not HHG. Maybe article can mention in broader developments around hybrid grid hierarchies. But topic is HHG; perhaps R3MG and polytopal agglomeration show neighboring agenda of hybrid hierarchical grids. Could say "Related multilevel work on polytopal meshes uses agglomeration hierarchies rather than classical HHG macro-refinement." This is a claim from data? "generate (hierarchies of) computational grids from an initial grid" from 24^^^^2query2^^^^4. Mark as related development. Maybe enough with 24^^^^2query2^^^^4 R3MG: fully automated, robust, dimension-independent, balanced and nested hierarchy, shape tightly close to axis aligned bounding boxes, efficient geometric multigrid [24^^^^2query2^^^^4]. Could be in this section as "Related agglomeration-based hierarchy construction." Need mark as related.
• Maybe also mention exact phrase "semi-structured solver framework" and "mixed structured/unstructured regions" as generalization.

Section6 applications and broader terminology

• Homogenized coefficients of divergence-form operators with random coefficients; laptop few seconds 2D, few minutes 3D moderate ellipticity contrast and few percentage points [^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5].
• geophysical simulations, coupled systems of PDEs, Stokes flow, mantle convection [^^^^2all:\2kℓ-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids2query2^^^^5].
• elasticostatics, incompressible Stokes flow, generalized Newtonian fluids [^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^8].
• H3-Mapping hierarchical hybrid representation for real-time dense mapping, quasi-heterogeneous feature grids, gradient-aided coverage-maximizing strategy, PSNR 35.92dB etc? Are those exact metrics from details. Could mention to illustrate broader use of term outside PDE. But article on HHGs maybe should cover this as broader terminology section. Need ensure section factual first. Perhaps phrase: "Outside finite-element and multigrid literature, a hierarchical hybrid representation has been used in real-time dense mapping. H3-Mapping combines a sparse voxel octree, multi-resolution hash grids, and quasi-heterogeneous feature grids; on Replica it reports 35.92dB PSNR and ^^^^2query2^^^^.^^^^2query2 average frame time (Jiang et al., 2024)." Since these claims in data.
• Perhaps also mention fixed grid plus hierarchical subdivision spatial index from (0705.0204) as another broader usage. Could say "A hybrid spatial indexing method begins with a fixed grid and recursively subdivides bins exceeding MaxBinRecords." But maybe too broad. Yet if section about broader terminology, including this is okay. But article should remain about HHG; could mention "Outside PDE solvers, related hierarchical hybrid grid ideas also appear..." and cite (0705.0204). Need once per section. This is okay. Need address misconception/controversies objectively maybe within conclusion-like paragraph? Could mention "The terminology is not uniform: in PDE computing, HHG usually denotes uniform refinement of coarse unstructured mesh; in robotics it denotes hierarchical hybrid representation; in spatial indexing it denotes a fixed grid plus hierarchical subdivision." This is inferential from papers. Mark "The terminology is not uniform." Is that claim from data? It is inferred, so need phrase "The literature suggests that the terminology is not uniform." Good.

Need maybe final concluding paragraph? Since article only, yes maybe end with a summary paragraph after section 6; but structure says 5-7 numbered h2 sections. Can include concluding paragraph after last section without heading maybe but likely okay. Yet task specified intro without heading and 5-7 sections. No explicit conclusion section unless one of numbered sections. Could make section 6 "Applications and terminological scope" with concluding note in paragraph.

Need ensure each section factual information first then context and significance. Can manage.

Let's craft detailed but concise. Need citations no more than once per section per paper. Let's ensure. Section^^^^2all:\2^^^^ cite (Kohl et al., 2018, Hannukainen et al., 2019). Section2 cite (&&&^^^^2all:\2^^^^8&&&). Section3 cite (&&&^^^^2all:\2^^^^8&&&, Mayr et al., 2021), maybe (Weinzierl et al., 2016) if including. Section4 cite (Kohl et al., 2018, &&&^^^^2all:\2^^^^8&&&, Drzisga et al., 2019). Section5 cite (&&&^^^^2query2^^^^&&&, &&&4^^^^2all:\2^^^^&&&), maybe (&&&^^^^2all:\2query2^^^^&&&). Section6 cite (Hannukainen et al., 2019, Jiang et al., 2024, 0705.0204). Could also (Kohl et al., 2018) but already used earlier no issue, but max once per section per paper.

Need not mention authors unless data mentioned. Some tasks requested reference authors and groups when mentioned in data. Data mentions Zander 2^^^^2query2all:\2^^^^5, 2^^^^2query2all:\2^^^^6; Parvizian, Duster, Jomo etc as references inside 2^^^^2query2all:\2query2^^^^ details. Maybe we can mention Zander in section 2 because multi-level hp-method uses overlay meshes (Zander 2^^^^2query2all:\2^^^^5, 2^^^^2query2all:\2^^^^6). However must cite arXiv IDs not those refs, but can mention "the multi-level PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^3-method (Zander 2^^^^2query2all:\2^^^^5, 2^^^^2query2all:\2^^^^6) uses overlay meshes..." from details. Since no arxiv ids for those, but if mention within sentence cite 2^^^^2query2all:\2query2^^^^ as source? probably okay. Similarly authors Godenschwager, Rüde? But not necessary.

Need maybe include equations. Section2 could include nested spaces and binary restriction. Section3 additive Schwarz, region decomposition. Need maybe include small table summarizing representative constructions. Could be in section ^^^^2all:\2^^^^ or 5. Let's see if necessary. Since content already paragraph heavy. Maybe a compact table in section ^^^^2all:\2^^^^ could clarify variants from data. Need keep brief. For example three approaches: Classical HHG in finite elements | unstructured coarse mesh + structured uniform refinement | papers Locally uniform partition | coarse simplices refined individually; reference simplex storage | ^^^^2all:\2H3-Mapping: Quasi-Heterogeneous Feature Grids for Real-time Dense Mapping Using Hierarchical Hybrid Representation2query2^^^^5 Semi-structured extension | structured and unstructured regions with conforming interfaces | 2^^^^2all:\2query2^^^^3 But table cell content brief. This is okay.

Need avoid using "this article". Start with paragraph.

Let's draft with precise wording. Hierarchical hybrid grids (HHGs) denote grid constructions that combine an unstructured coarse mesh with structured refinement, producing a multilevel, hierarchical mesh. In finite-element and multigrid settings, this hybrid grid offers the geometric flexibility of unstructured meshes with the computational efficiency typical for structured grids, while related formulations use an unstructured coarse grid to represent the variations of the coefficient field and then refine each coarse element in a “fully structured” manner, allowing very significant gains in memory usage (Kohl et al., 2018, Hannukainen et al., 2019).

^^^^2all:\2^^^^. Construction principles and canonical forms

Classical HHG approaches construct mesh hierarchies by regular refinement of a coarse unstructured mesh. A coarse, possibly unstructured, simplicial macro-triangulation PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^4 of the computational domain PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^5 is refined uniformly multiple times, with PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^6 for PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^7. Each macro element is refined uniformly, yielding regions where classical structured multigrid algorithms can be applied efficiently; most computational work is localized to the interiors of macro elements, while interfaces between macro elements retain their original unstructured nature (Mayr et al., 2021, Drzisga et al., 2019).

A closely related formulation starts from a coarse mesh of simplices, denoted PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^8, and refines each coarse element individually in a locally-uniform, structured way. The resulting fine mesh is a “locally uniform partition,”

PRESERVED_PLACEHOLDER_^^^^2all:\2^^^^9

where $h=2^{-l}H$^^^^2query2^^^^ is a reference refinement applied to all coarse elements. Only the fine mesh of the standard simplex is explicitly stored; the full fine mesh across the domain is reconstructed on-the-fly using affine mappings from this reference simplex to each coarse element (Hannukainen et al., 2019).

In software realizations, each coarse element is represented as a macro-primitive. Macro-faces, macro-edges, and macro-vertices together form a graph representing the mesh connectivity, and each macro-primitive acts both as a topological node and as a container for simulation data. Multiple refinement levels can be applied recursively, and the refined hierarchy provides the multiple mesh levels required by geometric multigrid (Kohl et al., 2018).

Construction	Defining feature	Representative source
Hierarchical hybrid grids	Unstructured coarse mesh with structured uniform refinement	(Kohl et al., 2018)
Locally uniform partition	Reference simplex refinement reconstructed on-the-fly	(Hannukainen et al., 2019)
Semi-structured generalization	Structured and unstructured regions with conforming interfaces	(Mayr et al., 2021)

These constructions establish the central compromise that recurs across the literature: geometric adaptability is retained at the macro level, while the fine-scale work is shifted to regular subgrids and regular kernels. A plausible implication is that HHG is best understood as a design pattern for discretization and solver organization rather than as a single mesh format.

2. Hierarchy of spaces, bases, and transfer operators

For immersed and high-order formulations, the hierarchy is expressed directly at the level of function spaces. Multigrid spaces form a sequence of nested function spaces,

$h=2^{-l}H$^^^^2all:\2^^^^

with $h=2^{-l}H$2 the finest space and $h=2^{-l}H$3 the coarsest. In uniform grids, coarsening is performed by successively reducing $h=2^{-l}H$4 down to $h=2^{-l}H$5. In multi-level $h=2^{-l}H$6-refined grids, the hierarchy is established first by reducing $h=2^{-l}H$7 on each overlay to minimum, then removing overlays, reducing the refinement level $h=2^{-l}H$8, so that each coarse space is a true subspace of the fine space (&&&^^^^2all:\2^^^^8&&&).

This construction depends on hierarchical basis functions. The finite cell method employs integrated Legendre polynomials as shape functions,

$h=2^{-l}H$9

and these basis functions are hierarchical in the sense that the basis functions for polynomial order $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$^^^^2query2^^^^ include all lower-order basis functions. The multi-level $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$^^^^2all:\2^^^^-method uses overlay meshes for local $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$2-refinement, preserving hierarchy among active basis functions; the data block attributes this overlay-mesh approach to Zander 2^^^^2query2all:\2^^^^5 and 2^^^^2query2all:\2^^^^6 (&&&^^^^2all:\2^^^^8&&&).

The hierarchical basis yields particularly simple intergrid transfer. If

$$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$3

then restriction is performed by simply dropping the degrees of freedom not present at the coarser level,

$$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$4

while prolongation is the transpose operation. All such operators are binary selection matrices; no explicit assembly is required. The system matrix inherits the hierarchical block structure,

$$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$5

so coarse-level data structures can be reused from the fine problem (&&&^^^^2all:\2^^^^8&&&).

The significance of this representation is methodological. The hierarchy is not only a sequence of meshes; it is simultaneously a hierarchy of basis functions, degrees of freedom, and linear operators. This suggests why HHG-related methods repeatedly emphasize simple prolongation, cheap restriction, and reuse of fine-level data structures.

3. Multigrid formulations and smoothing strategies

A recurring result in the HHG literature is that the grid hierarchy is designed as a solver hierarchy. In the finite cell setting, the solver uses a classical V-cycle multigrid with $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$6 pre-smoothing steps, residual computation, restriction to the coarse level, coarse-grid correction, prolongation, and $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$7 post-smoothing steps. The coarsest grid, usually $$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$8, is solved directly or with an iterative method (&&&^^^^2all:\2^^^^8&&&).

For cut-cell discretizations, standard smoothers are ineffective. The data block states that Jacobi and Gauss-Seidel are ineffective for FCM due to small or internally supported basis functions on cut elements. The replacement is additive Schwarz smoothing,

$$\mathcal{T}_h = lup(\mathcal{T}_H,\hat{\mathcal{T}})$$9

On uniform grids, the grouping is elementwise, taking all basis functions supported on an element. On multi-level $N_j(x)=\int_{-1}^x L_j(t)dt$^^^^2query2^^^^ grids, the grouping is patchwise, taking basis functions supported on a patch of base elements surrounding a node. Patchwise additive Schwarz smoothing yields convergence rates that are independent of mesh size $N_j(x)=\int_{-1}^x L_j(t)dt$^^^^2all:\2^^^^, independent of cut configuration, independent of polynomial order $N_j(x)=\int_{-1}^x L_j(t)dt$2 in 2D, and close to independent in 3D (&&&^^^^2all:\2^^^^8&&&).

The semi-structured generalization of HHG makes the same multigrid logic explicit in regional algebra. The domain is decomposed as

$N_j(x)=\int_{-1}^x L_j(t)dt$3

the global matrix is split as

$N_j(x)=\int_{-1}^x L_j(t)dt$4

and the transformation between composite and region representations is expressed by a Boolean matrix $N_j(x)=\int_{-1}^x L_j(t)dt$5,

$N_j(x)=\int_{-1}^x L_j(t)dt$6

The resulting region-oriented multigrid defines regional residuals and region Galerkin coarse operators,

$N_j(x)=\int_{-1}^x L_j(t)dt$7

and, under matching interface-stencil conditions, the method is mathematically equivalent to classical multigrid (Mayr et al., 2021).

A related operator-centric development is the quasi-matrix-free hybrid multigrid on dynamically adaptive Cartesian grids. There, the system matrix and transfer operators are not stored in full at every level; only hierarchical surpluses relative to geometric counterparts are kept,

$N_j(x)=\int_{-1}^x L_j(t)dt$8

while coarse-grid operators are formed by the Petrov-Galerkin relation

$N_j(x)=\int_{-1}^x L_j(t)dt$9

The paper explicitly characterizes this as a hybrid between geometric multigrid and algebraic multigrid, with almost matrix-free memory requirements for problems well-handled by geometric discretization (Weinzierl et al., 2016).

A common misconception is that the structured part of HHG automatically guarantees easy smoothing. The cut-cell results show the opposite: when the discretization creates severe local ill-conditioning, robust convergence depends on smoother design rather than on hierarchy alone.

4. Software architecture, matrix-free kernels, and scalability

HHG has also been developed as a software architecture. HyTeG is presented as a new generic higher-order finite-element framework for massively parallel simulations. Its modular architecture combines an unstructured topology with structured grid refinement, supports matrix-free multigrid implementations with excellent performance, and uses fully distributed data structures. The architecture separates topology, simulation data, and mesh-internal structure; it supports ghost layers, a three-layer communication design consisting of control, packing, and buffer layers, and graph partitioning with ParMETIS, Scotch, and Zoltan (Kohl et al., 2018).

Within each macro-primitive, regular subgrids enable stencil-based operator application and matrix-free matrix-vector multiplication. For constant-coefficient PDEs, one stencil can suffice per degree-of-freedom type inside a macro-primitive, and smoother kernels such as Jacobi, Gauss-Seidel, and Uzawa can be implemented efficiently in the stencil paradigm. The same hierarchy supports $V_0 \subset ...\subset V_\ell$^^^^2query2^^^^-multigrid, $V_0 \subset ...\subset V_\ell$^^^^2all:\2^^^^-multigrid, and $V_0 \subset ...\subset V_\ell$2-multigrid, because the degrees of freedom are organized modularly over vertices, edges, faces, and cells (Kohl et al., 2018).

Large-scale parallel implementations reinforce the same pattern. In the finite cell multigrid implementation, MPI and OpenMP are used together with distributed matrix libraries from Trilinos, specifically Epetra and AztecOO. Additive Schwarz smoothing is highly parallelizable because local block inversions are independent, and the hierarchical basis structure allows all coarse-level degrees of freedom to remain subsets of the fine-level degrees of freedom. The “popcorn benchmark” demonstrated excellent weak scaling up to 98,3^^^^2query2^^^^4 compute cores with 3.2 billion DOFs (&&&^^^^2all:\2^^^^8&&&).

For variable-coefficient vector-valued PDEs, stencil scaling on HHGs provides a matrix-free alternative to stored sparse matrices. The method is based on scaling constant reference stencils rather than evaluating the bilinear forms on-the-fly, and it is applied to linear elastostatics, incompressible Stokes flow, and a non-linear shear-thinning generalized Newtonian fluid. The abstract reports maximum speedups of 64% and ^^^^2all:\222^^^^% compared to the on-the-fly integration, and the largest considered example involved solving a Stokes problem with ^^^^2all:\222^^^^88 compute cores on the state of the art supercomputer SuperMUC-NG (Drzisga et al., 2019).

Taken together, these results indicate that HHG is as much about data movement and memory layout as about mesh topology. The literature repeatedly ties the success of HHG to cache-friendly kernels, halo exchange, matrix-free operator application, and the ability to keep most computation inside regular macro-element interiors.

5. Adaptivity and extensions beyond the classical HHG setting

The classical HHG construction is restrictive in one well-identified sense: traditional HHG approaches permit meshes derived from uniform refinement of a coarse mesh. Recent work therefore introduces adaptive variants that preserve the performance characteristics of block-structured fine grids while moving flexibility to the coarse level. The $V_0 \subset ...\subset V_\ell$3-refinement method performs classical, unstructured refinement only on the coarsest level of the hierarchy and keeps the number of structured refinement levels constant on the whole domain. Its hierarchy $V_0 \subset ...\subset V_\ell$4 is built by applying $V_0 \subset ...\subset V_\ell$5-refinement to the coarse grid and then $V_0 \subset ...\subset V_\ell$6 levels of structured regular refinement to all macro elements, and it uses a cheap error estimator derived from the sequence of approximations generated by the full multigrid scheme (&&&^^^^2query2^^^^&&&).

The corresponding estimators are written as

$V_0 \subset ...\subset V_\ell$7

The method is implemented in HyTeG, the time spent in AMR and in error estimation is described as a tiny percentage of total runtime, and the weak scaling studies extend to billions of DoFs (&&&^^^^2query2^^^^&&&).

A different extension replaces macro-refinement by agglomeration. R3MG performs agglomeration of polygonal and polyhedral grids using R-trees and states that the process is fully automated, robust, and dimension-independent; it automatically produces a balanced and nested hierarchy of agglomerates; and the shape of the agglomerates is tightly close to the respective axis aligned bounding boxes. Because the hierarchy is nested, geometric multigrid methods can be applied even when a hierarchy of grids is not present at construction time (&&&4^^^^2all:\2^^^^&&&).

On hybrid high-order discretizations, geometric multigrid has been extended to arbitrary polytopal agglomeration hierarchies in both two and three dimensions. The key ingredient is a modified skeleton space,

$V_0 \subset ...\subset V_\ell$8

which accommodates non-planar interfaces arising during coarsening while reducing the number of degrees of freedom. The paper proves convergence with respect to mesh size and number of levels, and reports that the approach extends naturally to other hybrid discretizations such as hybridizable discontinuous Galerkin and Weak Galerkin methods (&&&^^^^2all:\2query2^^^^&&&).

These developments suggest that the classical “uniform refinement of an unstructured coarse mesh” formulation is now one member of a larger family. The shared objective is stable multilevel structure with enough regularity for efficient kernels and enough flexibility for realistic geometry or adaptivity.

6. Applications and broader terminological scope

In numerical homogenization, HHGs are used to compute homogenized coefficients of divergence-form operators with random coefficients. The method combines a multiscale representation with a finite-element method on hierarchical hybrid grids, described as a semi-implicit method allowing for significant gains in memory usage and execution time. For moderate ellipticity contrast and for a precision of a few percentage points, the reported runtime is a few seconds in two dimensions and a few minutes in three dimensions on a laptop computer (Hannukainen et al., 2019).

In large-scale PDE simulation, HHG-based frameworks support coupled systems of PDEs and geophysical simulations such as Stokes flow and mantle convection, while stencil-scaling formulations target linear elastostatics, incompressible Stokes flow, and generalized Newtonian fluids (Kohl et al., 2018, Drzisga et al., 2019). In immersed analysis, hierarchical multigrid on uniform and multi-level $V_0 \subset ...\subset V_\ell$9-refined grids is applied to second-order problems arising from the Poisson equation and linear elasticity, and the numerical examples include perforated plates, a cube with spherical cavities, an aluminum rod, and the popcorn geometry benchmark (&&&^^^^2all:\2^^^^8&&&).

Outside finite-element and multigrid literature, related terminology appears in real-time dense mapping. H3-Mapping introduces a hierarchical hybrid representation that combines a sparse voxel octree for geometry with quasi-heterogeneous feature grids for texture. The texture model distinguishes rich-textured areas with low-frequency direction, weak-textured areas, and unstructured rich-textured areas; on Replica, the reported PSNR is 35.92dB and the average frame time is ^^^^2query2^^^^.^^^^2query2 (Jiang et al., 2024).

Related ideas also appear in spatial indexing. One hybrid spatial indexing method begins with a fixed grid, subdivides bins with too many records hierarchically, and pre-computes search queries for bins that do not contain any data records. In that setting, the hybridization is between a fixed grid approach and hierarchical subdivision, and the paper states that the method performs better than the quad tree if there are more divisions per layer (0705.0204).

The literature therefore suggests that the terminology is not uniform. In numerical PDEs, “hierarchical hybrid grids” usually denotes an unstructured coarse mesh with structured refinement and multilevel solver structure. In adjacent areas, the phrase “hierarchical hybrid representation” or related “hybrid grid” language can denote a different combination of hierarchy and local regularity. What remains consistent across these usages is the attempt to combine local structure with global flexibility.