paper.bib

@software{Holke_t8code_2022,
    author = {Holke, Johannes and Burstedde, Carsten and Knapp, David and Dreyer, Lukas and Elsweijer, Sandro and Uenlue, Veli and Markert, Johannes and Lilikakis, Ioannis and Boeing, Niklas},
    doi = {10.5281/zenodo.7034838},
    license = {GPL-2.0},
    month = {9},
    title = {{t8code}},
    url = {https://github.com/dlr-amr/t8code},
    version = {1.0.0},
    year = {2022}
}

@phdthesis{holke_scalable_2018,
  type = {PhD thesis},
  title = {Scalable algorithms for parallel tree-based adaptive mesh refinement with general element types},
  school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
  author = {Holke, Johannes},
  year = {2018},
  doi = {20.500.11811/7661}
}

@article{holke_optimized_2021,
    title = {An {Optimized}, {Parallel} {Computation} of the {Ghost} {Layer} for {Adaptive} {Hybrid} {Forest} {Meshes}},
    issn = {1064-8275},
    url = {https://epubs.siam.org/doi/abs/10.1137/20M1383033},
    doi = {10.1137/20M1383033},
    abstract = {We discuss parallel algorithms to compute the ghost layer in computational, distributed memory, recursively adapted meshes. Its creation is a fundamental, necessary task in executing most parallel, element-based computer simulations. Common methods differ in that the ghost layer may either be inherently part of the mesh data structure that is maintained and modified, or kept separate and constructed/deleted as needed. In this work, we present a design following the latter approach, which we chose for its modularity of algorithms and data structures. We target arbitrary adaptive, nonconforming forest-of-trees meshes of mixed element shapes, such as cubes, prisms, and tetrahedra, and restrict ourselves to ghost elements across mesh faces. Our algorithm has low code complexity and redundancy since we reduce it to generic codimension-1 subalgorithms that can be flexibly combined. We recover older algorithms for cubic elements as special cases and optimize further using recursive, amortized tree searches and traversals.},
    urldate = {2021-11-18},
    journal = {SIAM Journal on Scientific Computing},
    author = {Holke, Johannes and Knapp, David and Burstedde, Carsten},
    month = jan,
    year = {2021},
    keywords = {65D18, 65M50, 65Y05, 68W10, adaptive mesh refinement, forest of trees, ghost layer, parallel algorithms},
    pages = {C359--C385},
}

@article{geuzaine_gmsh_2009,
  title = {Gmsh: {A} 3-{D} finite element mesh generator with built-in pre- and post-processing facilities},
  volume = {79},
  issn = {1097-0207},
  shorttitle = {Gmsh},
  doi = {10.1002/nme.2579},
  abstract = {Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright © 2009 John Wiley \& Sons, Ltd.},
  language = {en},
  number = {11},
  urldate = {2021-12-01},
  journal = {International Journal for Numerical Methods in Engineering},
  author = {Geuzaine, Christophe and Remacle, Jean-François},
  year = {2009},
  keywords = {computer-aided design, finite element method, mesh generation, open-source software, post-processing},
  pages = {1309--1331},
}

@article{mengaldo_industry-relevant_2020,
  title = {Industry-{Relevant} {Implicit} {Large}-{Eddy} {Simulation} of a {High}-{Performance} {Road} {Car} via {Spectral}/hp {Element} {Methods}},
  volume = {abs/2009.10178},
  journal = {CoRR},
  author = {Mengaldo, Gianmarco and Moxey, David and Turner, Michael and Moura, Rodrigo C. and Jassim, Ayad and Taylor, Mark and Peiró, Joaquim and Sherwin, Spencer J.},
  year = {2020},
}

@incollection{koren_radial_2009,
  address = {Berlin, Heidelberg},
  title = {Radial {Basis} {Functions} for {Interface} {Interpolation} and {Mesh} {Deformation}},
  volume = {71},
  isbn = {9783642033438 9783642033445},
  urldate = {2022-01-14},
  booktitle = {Advanced {Computational} {Methods} in {Science} and {Engineering}},
  publisher = {Springer Berlin Heidelberg},
  author = {de Boer, A. and van Zuijlen, A. H. and Bijl, H.},
  editor = {Koren, Barry and Vuik, Kees},
  year = {2009},
  doi = {10.1007/978-3-642-03344-5_6},
  pages = {143--178},
}

@article{krais_flexi_2021,
  title = {{FLEXI}: {A} high order discontinuous {Galerkin} framework for hyperbolic–parabolic conservation laws},
  volume = {81},
  issn = {08981221},
  shorttitle = {{FLEXI}},
  doi = {10.1016/j.camwa.2020.05.004},
  language = {en},
  urldate = {2022-01-14},
  journal = {Computers \& Mathematics with Applications},
  author = {Krais, Nico and Beck, Andrea and Bolemann, Thomas and Frank, Hannes and Flad, David and Gassner, Gregor and Hindenlang, Florian and Hoffmann, Malte and Kuhn, Thomas and Sonntag, Matthias and Munz, Claus-Dieter},
  year = {2021},
  pages = {186--219},
}

@misc{noauthor_open_nodate,
  title = {Open {CASCADE} {Technology}},
  url = {https://www.opencascade.com/open-cascade-technology/},
  urldate = {2022-01-14},
  publisher = {Open Cascade},
}

@misc{juwels_fz_juelich,
  title = {JUWELS Supercomputer},
  author = {FZ Jülich},
  url = {https://www.fz-juelich.de/en/ias/jsc/systems/supercomputers/juwels},
  urldate = {2023-01-03},
  publisher = {FZ Jülich},
}

@misc{juqueen_fz_juelich,
  title = {JUQUEEN Supercomputer},
  author = {FZ Jülich},
  url = {https://hbp-hpc-platform.fz-juelich.de/?page_id=34},
  urldate = {2023-01-03},
  publisher = {FZ Jülich},
}


@techreport{elsweijer_curved_2021,
  title = {Curved {Domain} {Adaptive} {Mesh} {Refinement} with {Hexahedra}},
  url = {https://elib.dlr.de/143537/},
  abstract = {In tree-based adaptive mesh refinement (AMR) we store refinement
trees in the cells of an unstructured coarse mesh. This lets us combine the speed
and simpler management of structured refinement trees with the more flexible mesh
generation of the unstructured coarse mesh.
But this creates a conflict between performance and geometrical accuracy. If we
favor speed we reduce the cells in our coarse mesh and hence reduce the accuracy
of our geometrical representation. If we want more accurate results we generate a
finer coarse mesh and lose performance by managing more cells in our unstructured
coarse mesh.
To mitigate this conflict we present the prototype of an geometry description
which we implement in an already existing library. With this description we build
geometry adapted hexahedral refinement trees, which also support high-order curved
boundary cells. We also present examples on how to use this description.
Moreover, we test the speedup of this new algorithm compared with coarse meshes
with different geometrical errors.},
  language = {en},
  urldate = {2022-01-14},
  institution = {Hochschule Bonn-Rhein-Sieg},
  author = {Elsweijer, Sandro},
  month = jul,
  year = {2021},
}

@mastersthesis{elsweijer_evaluation_2022,
  author = {Elsweijer, Sandro},
  school = {Hochschule Bonn-Rhein-Sieg},
  year = {2022},
  title = {Evaluation and generic application scenarios for curved hexahedral adaptive mesh refinement},
  abstract = {In (dynamic) adaptive mesh refinement (AMR) an input mesh is refined or coarsened to the need of the numerical application. This refinement happens with no respect to the originally meshed domain and is therefore limited to the geometrical accuracy of the original input mesh. We presented a novel approach to equip this input mesh with additional geometry information, to allow refinement and high-order cells based on the geometry of the original domain.
We already showed a limited implementation of this algorithm. Now we evaluate this prototype with a numerical application and we prove its influence on the accuracy of certain numerical results. To be as practical as possible, we implement the ability to import meshes generated by Gmsh and equip them with the needed geometry information. Furthermore, we improve the mapping algorithm, which maps the geometry information of the boundary of a cell into the cell's volume.
With these preliminary steps done, we use out new approach in a simulation of the advection of a concentration along the boundary of a sphere shell and past the boundary of a rotating cylinder. We evaluate the accuracy of our approach in comparison to the conventional refinement of cells to answer our research question: How does the performance and accuracy of the hexahedral curved domain AMR algorithm compare to linear AMR when solving the advection equation with the linear finite volume method?
To answer this question, we show the influence of curved AMR on our simulation results and see, that it is even able to outperform far finer linear meshes in terms of accuracy. We also see that the current implementation of this approach is too slow for practical usage. We can therefore prove the benefits of curved AMR in certain, geometry-related application scenarios and show possible improvements to make it more feasible and practical in the future.},
  keywords = {adaptive mesh refinement, computational geometry, hexahedron, high-order meshes, mesh generation},
  url = {https://elib.dlr.de/186561/},
  doi = {10.13140/RG.2.2.34714.11203},
}

@article{burstedde_tetrahedral_2016,
  title = {A tetrahedral space-filling curve for nonconforming adaptive meshes},
  volume = {38},
  doi = {10.1137/15M1040049},
  journal = {SIAM Journal on Scientific Computing},
  author = {Burstedde, Carsten and Holke, Johannes},
  year = {2016},
  pages = {C471--C503},
}

@article{dey_towards_2001,
  title = {Towards curvilinear meshing in {3D}: the case of quadratic simplices},
  volume = {33},
  issn = {0010-4485},
  doi = {10.1016/S0010-4485(00)00120-2},
  abstract = {Issues related to curvilinear mesh generation are addressed. Curvilinear mesh geometry representation in the context of high-order finite element formulations and its impact on mesh validity is discussed. Specific criteria and associated mathematical relations are derived to ensure validity of meshes of quadratic simplices. An iterative algorithm for curving stright-edges meshes is described. Example curvilinear meshes of complex geometries obtained using this algorithm are included to show the validity of the presented approach.},
  number = {3},
  journal = {Computer-Aided Design},
  author = {Dey, S. and O'Bara, R. M. and Shephard, M. S.},
  year = {2001},
  keywords = {-Version analysis, Curvilinear mesh generation, Mesh generation},
  pages = {199--209},
}

@techreport{jacobs_characteristics_1933,
  title = {The characteristics of 78 related airfoil sections from tests in the variable-density wind tunnel},
  url = {https://ntrs.nasa.gov/citations/19930091108},
  number = {NACA Technical Report No. 460},
  urldate = {2021-07-19},
  author = {Jacobs, Eastman N and Ward, Kenneth E and Pinkerton, Robert M},
  year = {1933},
}

@article{berger_adaptive_1984,
  title = {Adaptive mesh refinement for hyperbolic partial differential equations},
  volume = {53},
  issn = {0021-9991},
  doi = {10.1016/0021-9991(84)90073-1},
  abstract = {An adaptive method based on the idea of multiple component grids for the solution of hyperbolic partial differential equations using finite difference techniques is presented. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. The approach is recursive in that fine grids can contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, error estimation procedure, and the data structures, and conclude with numerical examples in one and two space dimensions.},
  number = {3},
  journal = {Journal of Computational Physics},
  author = {Berger, Marsha J. and Oliger, Joseph},
  year = {1984},
  pages = {484--512},
}

@article{weinzierl_peano_2019,
  title = {The {Peano} {Software}-{Parallel}, {Automaton}-based, {Dynamically} {Adaptive} {Grid} {Traversals}},
  volume = {45},
  issn = {0098-3500},
  doi = {10.1145/3319797},
  number = {2},
  journal = {ACM Transactions on Mathematical Software},
  author = {Weinzierl, Tobias},
  year = {2019},
  pages = {1--41},
}

@misc{holke_t8code_2015,
  title = {t8code: {Parallel} {AMR} on hybrid non-conforming meshes},
  url = {https://github.com/holke/t8code},
  urldate = {2021-07-19},
  author = {Holke, Johannes and Burstedde, Carsten},
  year = {2015},
  keywords = {AMR, software},
}

@book{avila_paraview_2015,
  address = {Los Alamos},
  edition = {Full color version},
  title = {The {ParaView} guide: {Updated} for {ParaView} version 4.3},
  isbn = {978-1-930934-30-6},
  publisher = {Kitware},
  editor = {Avila, Lisa and Ayachit, Utkarsh},
  year = {2015},
}

@inproceedings{sundar_low-constant_2007,
  title = {Low-constant parallel algorithms for finite element simulations using linear octrees},
  isbn = {978-1-59593-764-3},
  doi = {10.1145/1362622.1362656},
  booktitle = {Proceedings of the 2007 {ACM}/{IEEE} conference on {Supercomputing} - {SC} '07},
  publisher = {ACM Press},
  author = {Sundar, Hari and Sampath, Rahul S. and Adavani, Santi S. and Davatzikos, Christos and Biros, George},
  editor = {Verastegui, Becky},
  year = {2007},
  pages = {1},
}

@article{muller_comparison_2013,
  title = {Comparison between adaptive and uniform discontinuous {Galerkin} simulations in dry {2D} bubble experiments},
  volume = {235},
  issn = {0021-9991},
  doi = {10.1016/j.jcp.2012.10.038},
  abstract = {Adaptive mesh refinement generally aims to increase computational efficiency without compromising the accuracy of the numerical solution. However it is an open question in which regions the spatial resolution can actually be coarsened without affecting the accuracy of the result. This question is investigated for a specific example of dry atmospheric convection, namely the simulation of warm air bubbles. For this purpose a novel numerical model is developed that is tailored towards this specific application. The compressible Euler equations are solved with a discontinuous Galerkin method. Time integration is done with an IMEX-method and the dynamic grid adaptivity uses space filling curves via the AMATOS function library. So far the model is able to simulate dry flow in two-dimensional geometry without subgrid-scale modeling. The model is tested with three standard test cases. An error indicator is introduced for a warm air bubble test case which allows one to compare the accuracy between different choices of refinement regions without knowing the exact solution. Essentially this is done by comparing features of the solution that are strongly sensitive to spatial resolution. For the rising warm air bubble the additional error by using adaptivity is smaller than 1\% of the total numerical error if the average number of elements used for the adaptive simulation is about a factor of two times smaller than the number used for the simulation with the uniform fine-resolution grid. Correspondingly the adaptive simulation is almost two times faster than the uniform simulation. Furthermore the adaptive simulation is more accurate than a uniform simulation when both use the same CPU-time.},
  journal = {Journal of Computational Physics},
  author = {Müller, Andreas and Behrens, Jörn and Giraldo, Francis X. and Wirth, Volkmar},
  year = {2013},
  keywords = {Adaptive mesh refinement, Discontinuous Galerkin, Dry warm air bubble, IMEX, Meteorology},
  pages = {371--393},
}

@article{klein_star_1999,
  title = {Star formation with 3-{D} adaptive mesh refinement: the collapse and fragmentation of molecular clouds},
  volume = {109},
  issn = {0377-0427},
  doi = {10.1016/S0377-0427(99)00156-9},
  abstract = {We describe a powerful methodology for numerical solution of 3-D self-gravitational hydrodynamics problems with unprecendented resolution. This code utilizes the technique of local adaptive mesh refinement (AMR), employing multiple grids at multiple levels of resolution. These grids are automatically and dynamically added and removed as necessary to maintain adequate resolution. This technology allows solution of problems that would be prohibitively expensive with methods using fixed resolution, and it is more versatile and efficient than competing methods of achieving variable resolution. The application of this technique to simulate the collapse and fragmentation of a molecular cloud, a key step in star formation is discussed. The simulation involves many orders of magnitude of variation in length scale as fragments form. In this paper we describe the methodology and present illustrative applications for both isothermal and nonisothermal cloud collapse. We describe the numerical Jeans condition, a new criterion for stability of self-gravitational gas dynamic problems. We find that the uniformly rotating, spherical clouds treated here first collapse to disks in the equatorial plane and then, in the presence of applied perturbations, form filamentary singularities that do not fragment while isothermal. As the collapse enters the non-isothermal phase, we show the evolutionary sequence that leads to the formation of a binary system consisting of protostellar cores surrounded by distinct protostellar disks. The scale of the disks, of order 100 AU, is consistent with observations of gaseous disks surrounding single T-Tauri stars and debris disks surrounding systems such as Beta Pictoris.},
  number = {1},
  journal = {Journal of Computational and Applied Mathematics},
  author = {Klein, Richard I.},
  year = {1999},
  keywords = {Gravitation, Hydrodynamics, ISM: clouds, Stars: formation},
  pages = {123--152},
}

@article{xie_generation_2013,
  title = {The generation of arbitrary order curved meshes for {3D} finite element analysis},
  volume = {51},
  issn = {1432-0924},
  doi = {10.1007/s00466-012-0736-4},
  abstract = {A procedure for generating curved meshes, suitable for high-order finite element analysis, is described. The strategy adopted is based upon curving a generated initial mesh with planar edges and faces by using a linear elasticity analogy. The analogy employs boundary loads that ensure that nodes representing curved boundaries lie on the true surface. Several examples, in both two and three dimensions, illustrate the performance of the proposed approach, with the quality of the generated meshes being analysed in terms of a distortion measure. The examples chosen involve geometries of particular interest to the computational fluid dynamics community, including anisotropic meshes for complex three dimensional configurations.},
  number = {3},
  journal = {Computational Mechanics},
  author = {Xie, Zhong Q. and Sevilla, Ruben and Hassan, Oubay and Morgan, Kenneth},
  month = mar,
  year = {2013},
  pages = {361--374},
}

@article{sundar_bottom-up_2008,
  title = {Bottom-{Up} {Construction} and 2:1 {Balance} {Refinement} of {Linear} {Octrees} in {Parallel}},
  volume = {30},
  doi = {10.1137/070681727},
  number = {5},
  journal = {SIAM Journal on Scientific Computing},
  author = {Sundar, Hari and Sampath, Rahul S. and Biros, George},
  year = {2008},
  pages = {2675--2708},
}

@article{burstedde_p4est_2011,
  title = {p4est: {Scalable} {Algorithms} for {Parallel} {Adaptive} {Mesh} {Refinement} on {Forests} of {Octrees}},
  volume = {33},
  doi = {10.1137/100791634},
  number = {3},
  journal = {SIAM Journal on Scientific Computing},
  author = {Burstedde, Carsten and Wilcox, Lucas C. and Ghattas, Omar},
  year = {2011},
  pages = {1103--1133},
}

@article{sherwin_mesh_2002,
  title = {Mesh generation in curvilinear domains using high-order elements},
  volume = {53},
  doi = {10.1002/nme.397},
  abstract = {Abstract The ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high-order algorithms in very complex geometries. Compact high-order algorithms such as spectral element or p-type finite element techniques offer the potential of high accuracy if the solution is smooth and a well-behaved mapping exists between the local sub-domains and a standard region. In this paper we address the problem of coupling high-order algorithms with unstructured mesh generation and CAD representation techniques. We present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings. The first strategy investigates the influence of an anisometric parametric surface representation on the quality of high-order unstructured meshes. We show that a straightforward splitting of a standard linear mesh for use in a high-order method can lead to the generation of distorted, possibly invalid, meshes even for simple computational domains such as cubes. The second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries. In the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher-order information about the surface into the mesh generation. Using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high-order discretization of the computational domain. Copyright © 2001 John Wiley \& Sons, Ltd.},
  number = {1},
  journal = {International Journal for Numerical Methods in Engineering},
  author = {Sherwin, S. J. and Peiró, J.},
  year = {2002},
  keywords = {curvature based adaption, high-order elements, high-order surface representation, hybrid meshing},
  pages = {207--223},
}

@article{shephard_adaptive_2005,
  title = {Adaptive mesh generation for curved domains},
  volume = {52},
  issn = {0168-9274},
  doi = {10.1016/j.apnum.2004.08.040},
  abstract = {This paper considers the technologies needed to support the creation of adaptively constructed meshes for general curved three-dimensional domains and outlines one set of solutions for providing them. A brief review of an effective way to integrate mesh generation/adaptation with CAD geometries is given. A set of procedures that support general h-adaptive refinement based on a mesh metric field is given. This is followed by examples that demonstrate the ability of the procedures to adaptively construct anisotropic meshes for flow problems. A procedure for the generation of strongly graded, curved meshes as needed for effective hp-adaptive simulations is also given.},
  number = {2},
  journal = {Applied Numerical Mathematics},
  author = {Shephard, Mark S. and Flaherty, Joseph E. and Jansen, Kenneth E. and Li, Xiangrong and Luo, Xiaojuan and Chevaugeon, Nicolas and Remacle, Jean-François and Beall, Mark W. and O'Bara, Robert M.},
  year = {2005},
  keywords = {Adaptive meshes, Anisotropic meshes, Curved meshes},
  pages = {251--271},
}

@article{luo_automatic_2004,
  title = {Automatic p-version mesh generation for curved domains},
  volume = {20},
  issn = {1435-5663},
  doi = {10.1007/s00366-004-0295-1},
  abstract = {To achieve the exponential rates of convergence possible with the p-version finite element method requires properly constructed meshes. In the case of piecewise smooth domains, these meshes are characterized by having large curved elements over smooth portions of the domain and geometrically graded curved elements to isolate the edge and vertex singularities that are of interest. This paper presents a procedure under development for the automatic generation of such meshes for general three-dimensional domains defined in solid modeling systems. Two key steps in the procedure are the determination of the singular model edges and vertices, and the creation of geometrically graded elements around those entities. The other key step is the use of general curved element mesh modification procedures to correct any invalid elements created by the curving of mesh entities on the model boundary, which is required to ensure a properly geometric approximation of the domain. Example meshes are included to demonstrate the features of the procedure.},
  number = {3},
  journal = {Engineering with Computers},
  author = {Luo, Xiao-Juan and Shephard, Mark S. and O’Bara, Robert M. and Nastasia, Rocco and Beall, Mark W.},
  month = sep,
  year = {2004},
  pages = {273--285},
}

@article{burstedde_coarse_2017,
  title = {Coarse {Mesh} {Partitioning} for {Tree}-{Based} {AMR}},
  volume = {Vol. 39},
  doi = {10.1137/16M1103518},
  journal = {SIAM Journal on Scientific Computing},
  author = {Burstedde, Carsten and Holke, Johannes},
  year = {2017},
  pages = {C364--C392},
}

@article{sahni_curved_2010,
  title = {Curved boundary layer meshing for adaptive viscous flow simulations},
  volume = {46},
  issn = {0168-874X},
  doi = {10.1016/j.finel.2009.06.016},
  abstract = {This paper presents an adaptive mesh control procedure suitable for use with high-order finite element methods to solve viscous flow problems. The procedure presented is an appropriate combination of anisotropic and boundary layer mesh adaptation that accounts for the need to use curved mesh edges and faces to maintain the required geometric approximation and mesh gradation within the boundary layers. The paper first discusses the mesh adaptation tools needed to create effective adapted meshes for higher-order viscous flow simulations. Consideration is then given to an overview of the individual mesh adaptation components that are combined to create such meshes. Finally, example results are given to demonstrate the importance of the techniques in accurate computation of physical quantities of interest and to also show the effectiveness of the developed procedures in dealing with domains of arbitrary geometric complexity.},
  number = {1},
  journal = {Finite Elements in Analysis and Design},
  author = {Sahni, O. and Luo, X. J. and Jansen, K. E. and Shephard, M. S.},
  year = {2010},
  keywords = {Boundary layer mesh, Higher-order analysis, Mesh adaptivity, Mesh curving, Viscous flow simulations},
  pages = {132--139},
}

@MASTERSTHESIS{Knapp17,
  Title                    = {{A}daptive {V}erfeinerung von {P}rismen},
  Author                   = {David Knapp},
  school     = {Rheinische Friedrich-Wilhelms-Universit{\"a}t Bonn},
  Type                     = {Bachelor's Thesis},
  Year                     = {2017}
}

@MASTERSTHESIS{Knapp20,
  Title                    = {A space-filling curve for pyramidal adaptive
                              mesh refinement},
  Author                   = {David Knapp},
  school     = {Rheinische Friedrich-Wilhelms-Universit{\"a}t Bonn},
  Type                     = {Master's Thesis},
  Year                     = {2020}
}

@mastersthesis{Becker_hanging_faces,
          author = {Florian Becker},
            year = {2021},
          school = {Universit{\"a}t zu K{\"o}ln},
           month = {Dezember},
           title = {Removing hanging faces from tree-based adaptive meshes for numerical simulations},
        keywords = {Adaptive Mesh Refinement
                    Space-filling curve
                    High-Performance Computing
                    Forest-of-trees
                    hanging nodes}
}

@mastersthesis{Lilikakis_removing,
           title = {Algorithms for tree-based adaptive meshes with incomplete trees},
            year = {2022},
          school = {Universit{\"a}t zu K{\"o}ln},
          author = {Ioannis Lilikakis},
             url = {https://elib.dlr.de/191968/},
        keywords = {AMR,
                    SFC,
                    Tree-based AMR,
                    Forest-of-trees,
                    Incomplete trees,
                    t8code,
                    HPC}
}

@mastersthesis{Dreyer2021,
           month = {Februar},
          school = {Rheinische Friedrich-Wilhems-Universit{\"a}t Bonn},
           title = {The local discontinuous galerkin method for the advection-diffusion equation on adaptive meshes},
          author = {Lukas Dreyer},
            note = {Erstgutachter: Prof. Dr. Carsten Burstedde, Zweitgutachter: Dr. Johannes Holke},
            year = {2021},
        keywords = {Adaptive mesh refinement, High-performance Computing, Discontinuous Galerkin, Advection, Diffusion},
             url = {https://elib.dlr.de/143969/}
}

@mastersthesis{Fussbroich_towards_2023,
            year = {2023},
          school = {Technische Hochschule K{\"o}ln},
           title = {Towards high-order, hybrid adaptive mesh refinement: Implementation and evaluation of curved unstructured mesh elements},
           month = {November},
          author = {Fu{\ss}broich, Jakob},
             url = {https://elib.dlr.de/200442/},
        keywords = {AMR, geometrical modelling, HPC, high-order meshes, tetrahedron}
}

@article{BursteddeSearch20,
author = {Burstedde, Carsten},
title = {Parallel Tree Algorithms for AMR and Non-Standard Data Access},
year = {2020},
issue_date = {December 2020},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {46},
number = {4},
issn = {0098-3500},
url = {https://doi.org/10.1145/3401990},
doi = {10.1145/3401990},
abstract = {We introduce several parallel algorithms operating on a distributed forest of adaptive quadtrees/octrees. They are targeted at large-scale applications relying on data layouts that are more complex than required for standard finite elements, such as hp-adaptive Galerkin methods, particle tracking and semi-Lagrangian schemes, and in-situ post-processing and visualization. Specifically, we design algorithms to derive an adapted worker forest based on sparse data, to identify owner processes in a top-down search of remote objects, and to allow for variable process counts and per-element data sizes in partitioning and parallel file I/O. We demonstrate the algorithms’ usability and performance in the context of a particle tracking example that we scale to 21e9 particles and 64Ki MPI processes on the Juqueen supercomputer, and we describe the construction of a parallel assembly of variably sized spheres in space creating up to 768e9 elements on the Juwels supercomputer.},
journal = {ACM Trans. Math. Softw.},
month = {nov},
articleno = {32},
numpages = {31},
keywords = {forest of octrees, particle tracking, Adaptive mesh refinement}
}

@article{TEUNISSEN2019106866,
title = {A geometric multigrid library for quadtree/octree AMR grids coupled to MPI-AMRVAC},
journal = {Computer Physics Communications},
volume = {245},
pages = {106866},
year = {2019},
issn = {0010-4655},
doi = {10.1016/j.cpc.2019.106866},
url = {https://www.sciencedirect.com/science/article/pii/S001046551930253X},
author = {J. Teunissen and R. Keppens},
keywords = {Multigrid, Elliptic solver, Octree, Adaptive mesh refinement, Divergence cleaning},
abstract = {We present an efficient MPI-parallel geometric multigrid library for quadtree (2D) or octree (3D) grids with adaptive refinement. Cartesian 2D/3D and cylindrical 2D geometries are supported, with second-order discretizations for the elliptic operators. Periodic, Dirichlet, and Neumann boundary conditions can be handled, as well as free-space boundary conditions for 3D Poisson problems, for which we use an FFT-based solver on the coarse grid. Scaling results up to 1792 cores are presented. The library can be used to extend adaptive mesh refinement frameworks with an elliptic solver, which we demonstrate by coupling it to MPI-AMRVAC. Several test cases are presented in which the multigrid routines are used to control the divergence of the magnetic field in magnetohydrodynamic simulations.}
}

@article{10.1145/1268776.1268779,
author = {Bangerth, W. and Hartmann, R. and Kanschat, G.},
title = {Deal.II—A General-Purpose Object-Oriented Finite Element Library},
year = {2007},
issue_date = {August 2007},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {33},
number = {4},
issn = {0098-3500},
url = {https://doi.org/10.1145/1268776.1268779},
doi = {10.1145/1268776.1268779},
abstract = {An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced object-oriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and application-specific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows us to avoid the computational costs frequently associated with abstract object-oriented class libraries.The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox.},
journal = {ACM Trans. Math. Softw.},
month = {aug},
pages = {24–es},
numpages = {27},
keywords = {software design, Object-orientation}
}



  
@Article{doi:10.1137/0733054,
  author   = {D\"{o}rfler, Willy},
  journal  = {SIAM Journal on Numerical Analysis},
  title    = {A Convergent Adaptive Algorithm for Poisson’s Equation},
  year     = {1996},
  number   = {3},
  pages    = {1106-1124},
  volume   = {33},
  abstract = {We construct a converging adaptive algorithm for linear elements applied to Poisson’s equation in two space dimensions. Starting from a macro triangulation, we describe how to construct an initial triangulation from a priori information. Then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions. It is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached. Extension to higher-order elements in two space dimension and numerical results are included.},
  doi      = {10.1137/0733054},
  eprint   = {https://doi.org/10.1137/0733054},
  url      = {https://doi.org/10.1137/0733054},
}







@article{doi:10.1137/0715049,
author = {Babuv\v{s}ka, I. and Rheinboldt, W. C.},
title = {Error Estimates for Adaptive Finite Element Computations},
journal = {SIAM Journal on Numerical Analysis},
volume = {15},
number = {4},
pages = {736-754},
year = {1978},
doi = {10.1137/0715049},

URL = { 
    
        https://doi.org/10.1137/0715049
    
    

},
eprint = { 
    
        https://doi.org/10.1137/0715049
    
    

}
,
    abstract = { A mathematical theory is developed for a class of a-posteriors error estimates of finite element solutions. It is based on a general formulation of the finite element method in terms of certain bilinear forms on suitable Hilbert spaces. The main theorem gives an error estimate in terms of localized quantities which can be computed approximately. The estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same. The theoretical results also lead to a heuristic characterization of optimal meshes, which in turn suggests a strategy for adaptive mesh refinement. Some numerical examples show the approach to be very effective. }
}


@misc{schlottkelakemper2020trixi,
  title={{T}rixi.jl: {A}daptive high-order numerical simulations
         of hyperbolic {PDE}s in {J}ulia},
  author={Schlottke-Lakemper, Michael and Gassner, Gregor J and
          Ranocha, Hendrik and Winters, Andrew R},
  year={2020},
  month={08},
  howpublished={\url{https://github.com/trixi-framework/Trixi.jl}},
  doi={10.5281/zenodo.3996439}
}

@ARTICLE{BursteddeWilcoxGhattas11,
  author = {Carsten Burstedde and Lucas C. Wilcox and Omar Ghattas},
  title = {{\texttt{p4est}}: Scalable Algorithms for Parallel Adaptive Mesh
           Refinement on Forests of Octrees},
  journal = {SIAM Journal on Scientific Computing},
  volume = {33},
  number = {3},
  pages = {1103-1133},
  year = {2011},
  doi = {10.1137/100791634}
}

@article{libMeshPaper,
  author = {Benjamin S. Kirk and John W. Peterson and Roy H. Stogner and Graham F. Carey},
  title = {{\texttt{libMesh}: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations}},
  journal = {Engineering with Computers},
  volume = 22,
  number = {3--4},
  pages = {237--254},
  year = 2006,
  note = {\url{https://doi.org/10.1007/s00366-006-0049-3}}
}

@article{macneice2000paramesh,
  title={PARAMESH: A parallel adaptive mesh refinement community toolkit},
  author={MacNeice, Peter and Olson, Kevin M and Mobarry, Clark and De Fainchtein, Rosalinda and Packer, Charles},
  journal={Computer physics communications},
  volume={126},
  number={3},
  pages={330--354},
  year={2000},
  publisher={Elsevier}
}

@techreport{gunney2013scalable,
  title={Scalable mesh management for patch-based amr},
  author={Gunney, BN},
  year={2013},
  institution={Lawrence Livermore National Lab.(LLNL), Livermore, CA (United States)}
}

@article{holke2023t8code,
  title={t8code v. 1.0-modular adaptive mesh refinement in the exascale era},
  author={Holke, Johannes and Burstedde, Carsten and Knapp, David and Dreyer, Lukas and Elsweijer, Sandro and {\"U}nl{\"u}, Veli and Markert, Johannes and Lilikakis, Ioannis and B{\"o}ing, Niklas and Ponnusamy, Prasanna and others},
  year={2023}
}

@article{geuzaine2009gmsh,
  title={Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities},
  author={Geuzaine, Christophe and Remacle, Jean-Fran{\c{c}}ois},
  journal={International journal for numerical methods in engineering},
  volume={79},
  number={11},
  pages={1309--1331},
  year={2009},
  publisher={Wiley Online Library}
}