×

zbMATH — the first resource for mathematics

The Dune framework: basic concepts and recent developments. (English) Zbl 07288707
Summary: This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state [P. Bastian et al., Computing 82, No. 2–3, 103–119 (2008; Zbl 1151.65089); Computing 82, No. 2–3, 121–138 (2008; Zbl 1151.65088)]. This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.

MSC:
65-XX Numerical analysis
68-XX Computer science
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. part I: Abstract framework, Computing, 82, 2-3, 103-119 (2008) · Zbl 1151.65089
[2] Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klöfkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O., A generic grid interface for parallel and adaptive scientific computing. part II: Implementation and tests in DUNE, Computing, 82, 2-3, 121-138 (2008) · Zbl 1151.65088
[3] Vey, S.; Voigt, A., AMDiS: adaptive multidimensional simulations, Comput. Vis. Sci., 10, 1, 57-67 (2007)
[4] Arndt, D.; Bangerth, W.; Clevenger, T. C.; Davydov, D.; Fehling, M.; Garcia-Sanchez, D.; Harper, G.; Heister, T.; Heltai, L.; Kronbichler, M.; Kynch, R. M.; Maier, M.; Pelteret, J.-P.; Turcksin, B.; Wells, D., The deal.ii library, version 9.1, J. Numer. Math. (2019), online-first. · Zbl 1435.65010
[5] Logg, A.; Mardal, K.-A.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012), Springer Publishing Company, Incorporated · Zbl 1247.65105
[6] Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 3-4, 251-265 (2012), URL https://freefem.org/ · Zbl 1266.68090
[7] Gawlok, S.; Gerstner, P.; Haupt, S.; Heuveline, V.; Kratzke, J.; Lösel, P.; Mang, K.; Schmidtobreick, M.; Schoch, N.; Schween, N.; Schwegler, J.; Song, C.; Wlotzka, M., HiFlow3 - Technical Report on Release 2.0Preprint Series of the Engineering Mathematics and Computing Lab (EMCL) 0 (06) (2017)
[8] Liu, Q.; Mo, Z.; Zhang, A.; Yang, Z., JAUMIN: a programming framework for large-scale numerical simulation on unstructured meshes, CCF Trans. High Perform. Comput., 1, 1, 35-48 (2019)
[9] Kolev, T.; Dobrev, V., MFEM: Modular finite element methods library (2010)
[10] Netgen/NGSolve: high performance multiphysics finite element software, https://ngsolve.org/.
[11] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Dener, A.; Eijkhout, V.; Gropp, W. D.; Karpeyev, D.; Kaushik, D.; Knepley, M. G.; May, D. A.; McInnes, L. C.; Mills, R. T.; Munson, T.; Rupp, K.; Sanan, P.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H., PETSc Web page (2019), URL https://www.mcs.anl.gov/petsc
[12] Vogel, A.; Reiter, S.; Rupp, M.; Nägel, A.; Wittum, G., UG 4: A novel flexible software system for simulating PDE based models on high performance computers, Comput. Vis. Sci., 16, 4, 165-179 (2013) · Zbl 1375.35003
[13] Brooks, F., The Mythical Man-Month: Essays on Software Engineering (1975), Addison-Wesley, URL https://archive.org/details/mythicalmanmonth00fred
[14] Rasmussen, A. F.; Sandve, T. H.; Bao, K.; Lauser, A.; Hove, J.; Skaflestad, B.; Klöfkorn, R.; Blatt, M.; Rustad, A. B.; Sævareid, O., The open porous media flow reservoir simulator, Comput. Math. Appl. (2020)
[15] Koch, T.; Gläser, D.; Weishaupt, K.; Ackermann, S.; Beck, M.; Becker, B.; Burbulla, S.; Class, H.; Coltman, E.; Emmert, S.; Fetzer, T.; Grüninger, C.; Heck, K.; Hommel, J.; Kurz, T.; Lipp, M.; Mohammadi, F.; Scherrer, S.; Schneider, M.; Seitz, G.; Stadler, L.; Utz, M.; Weinhardt, F.; Flemisch, B., Dumux 3 - an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling, Comput. Math. Appl. (2020)
[16] Götschel, S.; Weiser, M.; Schiela, A., Solving optimal control problems with the Kaskade 7 finite element toolbox, (Advances in DUNE (2012), Springer), 101-112
[17] Drohmann, M.; Haasdonk, B.; Kaulmann, S.; Ohlberger, M., A software framework for reduced basis methods using dune-RB and rbmatlab, (Dedner, A.; Flemisch, B.; Klöfkorn, R., Advances in DUNE (2012), Springer), 77-88
[18] Lee, C. T.; Moody, J. B.; Amaro, R. E.; McCammon, J. A.; Holst, M. J., The implementation of the colored abstract simplicial complex and its application to mesh generation, ACM Trans. Math. Software, 45, 3 (2019) · Zbl 07193377
[19] Ainsworth, M.; Oden, J., A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142, 1, 1-88 (1997) · Zbl 0895.76040
[20] Bastian, P.; Engwer, C.; Göddeke, D.; Iliev, O.; Ippisch, O.; Ohlberger, M.; Turek, S.; Fahlke, J.; Kaulmann, S.; Müthing, S.; Ribbrock, D., EXA-DUNE: Flexible PDE solvers, numerical methods and applications, (Lopes, A.; etal., Euro-Par 2014: Parallel Processing Workshops. Euro-Par 2014 International Workshops, Porto, Portugal, August 25-26, 2014, Revised Selected Papers, Part II.. Euro-Par 2014: Parallel Processing Workshops. Euro-Par 2014 International Workshops, Porto, Portugal, August 25-26, 2014, Revised Selected Papers, Part II., Lecture Notes in Computer Science, vol. 8806 (2014), Springer), 530-541
[21] Bastian, P.; Engwer, C.; Fahlke, J.; Geveler, M.; Göddeke, D.; Iliev, O.; Ippisch, O.; Milk, R.; Mohring, J.; Müthing, S.; Ohlberger, M.; Ribbrock, D.; Turek, S., Hardware-based efficiency advances in the EXA-DUNE project, (Software for Exascale Computing - SPPEXA 2013-2015. Software for Exascale Computing - SPPEXA 2013-2015, Lecture Notes in Computational Science and Engineering (2016), Springer Verlag), 3-23
[22] Klöfkorn, R., Efficient matrix-free implementation of discontinuous Galerkin methods for compressible flow problems, (Handlovicova, A.; etal., Proceedings of the ALGORITMY (2012)), 11-21, URL http://www.iam.fmph.uniba.sk/algoritmy2012/zbornik/2Kloefkornf.pdf · Zbl 1278.35157
[23] Schmidt, A.; Siebert, K., Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA (2005), Springer, URL http://www.alberta-fem.de/ · Zbl 1068.65138
[24] Alkämper, M.; Dedner, A.; Klöfkorn, R.; Nolte, M., The DUNE-ALUGrid module, Arch. Numer. Softw., 4, 1, 1-28 (2016)
[25] Fomins, A.; Oswald, B., Dune-CurvilinearGrid: Parallel dune grid manager for unstructured Tetrahedral curvilinear meshes (2016), arXiv e-prints arXiv:1612.02967
[26] Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181
[27] Sander, O.; Koch, T.; Schröder, N.; Flemisch, B., The Dune FoamGrid implementation for surface and network grids, Arch. Numer. Softw., 5, 1, 217-244 (2017)
[28] Bastian, P.; Birken, K.; Johannsen, K.; Lang, S.; Neuß, N.; Rentz-Reichert, H.; Wieners, C., UG - A flexible software toolbox for solving partial differential equations, Comput. Vis. Sci., 1, 1, 27-40 (1997) · Zbl 0970.65129
[29] Gersbacher, C., The Dune-PrismGrid module, (Dedner, A.; Flemisch, B.; Klöfkorn, R., Advances in DUNE (2012)), 33-44
[30] Bastian, P.; Buse, G.; Sander, O., Infrastructure for the coupling of dune grids, (Kreiss, G.; Lötstedt, P.; Målqvist, A.; Neytcheva, M., Numerical Mathematics and Advanced Applications 2009 (2010), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 107-114 · Zbl 1311.76100
[31] Engwer, C.; Müthing, S., Concepts for flexible parallel multi-domain simulations, (Domain Decomposition Methods in Science and Engineering XXII (2016), Springer), 187-195 · Zbl 1338.65262
[32] Müthing, S., A flexible framework for multi physics and multi domain PDE simulations (2015), Universität Stuttgart, (Ph.D. thesis)
[33] Gräser, C.; Sander, O., The dune-subgrid module and some applications, Computing, 86, 4, 269 (2009) · Zbl 1179.65154
[34] Blatt, M.; Burchardt, A.; Dedner, A.; Engwer, C.; Fahlke, J.; Flemisch, B.; Gersbacher, C.; Gräser, C.; Gruber, F.; Grüninger, C.; Kempf, D.; Klöfkorn, R.; Malkmus, T.; Müthing, S.; Nolte, M.; Piatkowski, M.; Sander, O., The distributed and unified numerics environment, Version 2.4, Arch. Numer. Softw., 4, 100, 13-29 (2016)
[35] Klöfkorn, R.; Nolte, M., Performance pitfalls in the dune grid interface, (Dedner, A.; Flemisch, B.; Klöfkorn, R., Advances in DUNE (2012), Springer Berlin Heidelberg), 45-58
[36] Elman, H.; Silvester, D.; Wathen, A., Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics (2014), Oxford University Press · Zbl 1304.76002
[37] Blatt, M.; Bastian, P., The iterative solver template library, (Kågström, B.; Elmroth, E.; Dongarra, J.; Waśniewski, J., Applied Parallel Computing. State of the Art in Scientific Computing. Applied Parallel Computing. State of the Art in Scientific Computing, Lecture Notes in Computer Science, vol. 4699 (2007), Springer), 666-675
[38] Blatt, M.; Bastian, P., On the generic parallelisation of iterative solvers for the finite element method, Int. J. Comput. Sci. Engrg., 4, 1, 56-69 (2008)
[39] Blatt, M., A parallel algebraic multigrid method for elliptic problems with highly discontinuous coefficients (2010), Universtität Heidelberg, (Ph.D. thesis) · Zbl 1194.65002
[40] Bastian, P.; Blatt, M.; Scheichl, R., Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems, Numer. Linear Algebra Appl., 2, 19, 367-388 (2012) · Zbl 1274.65313
[41] Ippisch, O.; Blatt, M., Scalability test of \(\mu\phi\) and the parallel algebraic multigrid solver of dune-istl, (JÜLich Blue Gene/P Extreme Scaling Workshop, No. FZJ-JSC-IB-2011-02. JÜLich Supercomputing Centre (2011)), 21-26, doi: 2128/7309
[42] Yang, U. M., On the use of relaxation parameters in hybrid smoothers, Numer. Linear Algebra Appl., 11, 2-3, 155-172 (2004) · Zbl 1164.65361
[43] Kretz, M., Extending C++ for explicit data-parallel programming via SIMD vector types (2015), Goethe University Frankfurt am Main, (Ph.D. thesis)
[44] Kretz, M.; Lindenstruth, V., Vc: A c++ library for explicit vectorization, Softw. - Pract. Exp., 42, 11, 1409-1430 (2012)
[45] Fog, A., C++ vector class library (2013), URL http://www.agner.org/optimize/vectorclass.pdf
[46] Klöfkorn, R.; Kvashchuk, A.; Nolte, M., Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids, Comput. Geosci., 21, 5, 909-919 (2017) · Zbl 1396.76057
[47] Dedner, A.; Müller, E.; Scheichl, R., Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio, Internat. J. Numer. Methods Fluids, 80, 1, 76-102 (2016)
[48] Ciarlet, P. G., The finite element method for elliptic problems, Vol. 40 (2002), SIAM
[49] Engwer, C.; Gräser, C.; Müthing, S.; Sander, O., Function space bases in the dune-functions module (2018), ArXiv e-prints arXiv:1806.09545
[50] Engwer, C.; Gräser, C.; Müthing, S.; Sander, O., The interface for functions in the dune-functions module, Arch. Numer. Softw., 5, 1, 95-109 (2017), arXiv:1512.06136
[51] Dalcin, L. D.; Paz, R. R.; Kler, P. A.; Cosimo, A., Parallel distributed computing using python, Adv. Water Resour., 34, 9, 1124-1139 (2011)
[52] Rathgeber, F.; Ham, D. A.; Mitchell, L.; Lange, M.; Luporini, F.; McRae, A. T.T.; Bercea, G.-T.; Markall, G. R.; Kelly, P. H.J., Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software, 43, 3, 24:1-24:27 (2016) · Zbl 1396.65144
[53] Jakob, W.; Rhinelander, J.; Moldovan, D., Pybind11 - Seamless operability between C++11 and Python (2017), URL https://github.com/pybind/pybind11
[54] Dedner, A.; Nolte, M., The Dune Python Module (2018), ArXiv e-prints arXiv:1807.05252
[55] Alnæs, M. S.; Logg, A.; Ølgaard, K. B.; Rognes, M. E.; Wells, G. N., Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Software, 40, 2, 9:1-9:37 (2014) · Zbl 1308.65175
[56] Kempf, D.; Koch, T., System testing in scientific numerical software frameworks using the example of DUNE, Arch. Numer. Softw., 5, 1, 151-168 (2017)
[57] Burstedde, C.; Wilcox, L.; Ghattas, O., P4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM J. Sci. Comput., 33, 3, 1103-1133 (2011) · Zbl 1230.65106
[58] Badia, S.; Martín, A. F.; Principe, J., FEMPAR: An object-oriented parallel finite element framework, Arch. Comput. Methods Eng., 25, 2, 195-271 (2018) · Zbl 1392.65005
[59] Xie, T.; Seol, S.; Shephard, M., Generic components for petascale adaptive unstructured mesh-based simulations, Eng. Comput., 30, 1, 79-95 (2014)
[60] Kirk, B.; Peterson, J.; Stogne, R.; Carey, G., Libmesh: A c++ library for parallel adaptive mesh refinement/coarsening simulations, Eng. Comput., 22, 3-4, 237-254 (2006)
[61] Dedner, A.; Klöfkorn, R., A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems, J. Sci. Comput., 47, 3, 365-388 (2011) · Zbl 1229.65175
[62] Schuster, D.; Brdar, S.; Baldauf, M.; Dedner, A.; Klöfkorn, R.; Kröner, D., On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture, Meteorol. Z., 23, 4, 449-464 (2014)
[63] Dedner, A.; Klöfkorn, R.; Kränkel, M., Continuous finite-elements on non-conforming grids using discontinuous Galerkin stabilization, (Fuhrmann, J.; etal., Finite Volumes for Complex Applications VII. Finite Volumes for Complex Applications VII, Springer Proceedings in Mathematics & Statistics, vol. 77 (2014), Springer), 207-215 · Zbl 1298.65169
[64] Dedner, A.; Kane, B.; Klöfkorn, R.; Nolte, M., Python framework for hp-adaptive discontinuous Galerkin methods for two-phase flow in porous media, Appl. Math. Model., 67, 179-200 (2019) · Zbl 07183423
[65] Kane, B.; Klöfkorn, R.; Gersbacher, C., Hp-adaptive discontinuous Galerkin methods for porous media flow, (Cancès, C.; Omnes, P., Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems: FVCA 8, Lille, France, June 2017 (2017), Springer International Publishing: Springer International Publishing Cham), 447-456 · Zbl 1365.76159
[66] Kane, B., Adaptive higher order discontinuous Galerkin methods for porous-media multi-phase flow with strong heterogeneities (2018), Universität Stuttgart, (Dissertation)
[67] Klöfkorn, R.; Kröner, D.; Ohlberger, M., Parallel adaptive simulation of PEM fuel cells, (Krebs, H.-J.; Jäger, W., Mathematics - Key Technology for the Future (2008), Springer), 235-249
[68] Gersbacher, C., Higher-order discontinuous finite element methods and dynamic model adaptation for hyperbolic systems of conservation laws (2017), Albert-Ludwigs Universität Freiburg, (Dissertation) · Zbl 1370.65052
[69] Gräser, C.; Kornhuber, R.; Sack, U., Numerical simulation of coarsening in binary solder alloys, Comput. Mater. Sci., 93, 221-233 (2014)
[70] Dedner, A.; Klöfkorn, R.; Nolte, M., Python Bindings for the DUNE-FEM module (2020), Zenodo
[71] Barkley, D., A model for fast computer simulation of waves in excitable media, Physica, 49, 61-70 (1991)
[72] Alkämper, M.; Gaspoz, F.; Klöfkorn, R., A weak compatibility condition for newest vertex bisection in any dimension, SIAM J. Sci. Comput., 40, 6, A3853-A3872 (2018) · Zbl 1404.65247
[73] Alkämper, M.; Klöfkorn, R., Distributed newest vertex bisection, J. Parallel Distrib. Comput., 104, 1-11 (2017)
[74] Deckelnick, K.; Dziuk, G.; Elliott, C. M., Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14, 139-232 (2005) · Zbl 1113.65097
[75] Klöfkorn, R.; Nolte, M., Solving the reactive compressible Navier-Stokes equations in a moving domain, (Binder, K.; Münster, G.; Kremer, M., NIC Symposium 2014 - Proceedings, Vol. 47 (2014), John von Neumann Institute for Computing Jülich), 353-362, doi: 2128/5919
[76] Bernardi, C.; Maday, Y.; Patera, A., Domain decomposition by the mortar element method, (Kaper, H.; Garbey, M.; Pieper, G., Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters. Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol. 384 (1993), Springer), 269-286 · Zbl 0799.65124
[77] Becker, R.; Hansbo, P.; Stenberg, R., A finite element method for domain decomposition with non-matching grids, ESAIM Math. Model. Numer. Anal., 37, 2, 209-225 (2003) · Zbl 1047.65099
[78] Lazarov, R. D.; Pasciak, J. E.; Schöberl, J.; Vassilevski, P. S., Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids, Numer. Math., 96, 2, 295-315 (2003) · Zbl 1095.65103
[79] Gander, M. J.; Japhet, C.; Maday, Y.; Nataf, F., A new cement to glue nonconforming grids with robin interface conditions: the finite element case, (Domain Decomposition Methods in Science and Engineering (2005), Springer), 259-266 · Zbl 1067.65144
[80] Bastian, P.; Engwer, C., An unfitted finite element method using discontinuous Galerkin, Internat. J. Numer. Methods Engrg., 79, 1557-1576 (2009) · Zbl 1176.65131
[81] Engwer, C.; Heimann, F., Dune-UDG: A cut-cell framework for unfitted discontinuous Galerkin methods, (Advances in DUNE (2012), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 89-100
[82] Burman, E.; Claus, S.; Hansbo, P.; Larson, M. G.; Massing, A., CutFEM: Discretizing geometry and partial differential equations, Intern. J. Numer. Methods Engrg., 104, 472-501 (2015) · Zbl 1352.65604
[83] Koch, T.; Heck, K.; Schröder, N.; Class, H.; Helmig, R., A new simulation framework for soil-root interaction, evaporation, root growth, and solute transport, Vadose Zone J. (2018)
[84] Gander, M. J.; Japhet, C., Algorithm 932: PANG: software for nonmatching grid projections in 2d and 3D with linear complexity, ACM Trans. Math. Softw., 40, 1, 6 (2013) · Zbl 1295.65120
[85] Engwer, C.; Nüßing, A., Geometric reconstruction of implicitly defined surfaces and domains with topological guarantees, ACM Trans. Math. Softw., 44, 2, 14 (2017) · Zbl 06920076
[86] Gräser, C.; Kornhuber, R., Multigrid methods for obstacle problems, J. Comput. Math., 27, 1, 1-44 (2009) · Zbl 1199.65401
[87] Gräser, C.; Sander, O., Truncated nonsmooth Newton multigrid methods for block-separable minimization problems, IMA J. Numer. Anal., 39, 1, 454-481 (2019) · Zbl 07208111
[88] Han, W.; Reddy, B. D., Plasticity (2013), Springer
[89] Sander, O., Solving primal plasticity increment problems in the time of a single predictor-corrector iteration (2017), ArXiv e-prints arXiv:1707.03733
[90] Alberty, J.; Carstensen, C.; Zarrabi, D., Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg., 171, 175-204 (1999) · Zbl 0956.74049
[91] Neff, P.; Sydow, A.; Wieners, C., Numerical approximation of incremental infinitesimal gradient plasticity, Internat. J. Numer. Methods Engrg., 77, 414-436 (2009) · Zbl 1155.74316
[92] Hou, T. Y.; Wu, X., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 1, 169-189 (1997) · Zbl 0880.73065
[93] Efendiev, Y.; Hou, T. Y., Multiscale finite element methods, (Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4 (2009), Springer: Springer New York), xii+234, Theory and applications · Zbl 1163.65080
[94] Henning, P.; Ohlberger, M.; Schweizer, B., An adaptive multiscale finite element method, Multiscale Model. Simul., 12, 3, 1078-1107 (2014) · Zbl 1312.65191
[95] E, W.; Engquist, B., The heterogeneous multiscale methods, Commun. Math. Sci., 1, 1, 87-132 (2003) · Zbl 1093.35012
[96] Ohlberger, M., A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model. Simul., 4, 1, 88-114 (2005) · Zbl 1090.65128
[97] Abdulle, A., On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul., 4, 2, 447-459 (2005) · Zbl 1092.65093
[98] Hughes, T. J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127, 1-4, 387-401 (1995) · Zbl 0866.76044
[99] Hughes, T. J.R.; Feijóo, G. R.; Mazzei, L.; Quincy, J.-B., The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166, 1-2, 3-24 (1998) · Zbl 1017.65525
[100] Larson, M. G.; Malqvist, A., Adaptive variational multiscale methods based on a posteriori error estimation: duality techniques for elliptic problems, (Multiscale Methods in Science and Engineering. Multiscale Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng., vol. 44 (2005), Springer: Springer Berlin), 181-193 · Zbl 1105.65353
[101] Malqvist, A.; Peterseim, D., Localization of elliptic multiscale problems, Math. Comp., 83, 290, 2583-2603 (2014) · Zbl 1301.65123
[102] Henning, P.; Malqvist, A.; Peterseim, D., A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal., 48, 5, 1331-1349 (2014) · Zbl 1300.35011
[103] Engwer, C.; Henning, P.; Målqvist, A.; Peterseim, D., Efficient implementation of the localized orthogonal decomposition method, Comput. Methods Appl. Mech. Engrg., 350, 123-153 (2019) · Zbl 1441.65100
[104] Albrecht, F.; Haasdonk, B.; Kaulmann, S.; Ohlberger, M., The localized reduced basis multiscale method, Proc. ALGORITMY, 393-403 (2012) · Zbl 1278.65172
[105] Ohlberger, M.; Schindler, F., Error control for the localized reduced basis multiscale method with adaptive on-line enrichment, SIAM J. Sci. Comput., 37, 6, A2865-A2895 (2015) · Zbl 1329.65255
[106] Ohlberger, M.; Rave, S.; Schindler, F., True error control for the localized reduced basis method for parabolic problems, (Model Reduction of Parametrized Systems. Model Reduction of Parametrized Systems, MS&A. Model. Simul. Appl., vol. 17 (2017), Springer, Cham), 169-182 · Zbl 1448.65170
[107] Efendiev, Y.; Galvis, J.; Hou, T. Y., Generalized multiscale finite element methods (gmsfem), J. Comput. Phys., 251, 116-135 (2013) · Zbl 1349.65617
[108] Chung, E. T.; Efendiev, Y.; Li, G., An adaptive GMsFEM for high-contrast flow problems, J. Comput. Phys., 273, 54-76 (2014) · Zbl 1354.65242
[109] Chung, E. T.; Efendiev, Y.; Leung, W. T., An adaptive generalized multiscale discontinuous Galerkin method for high-contrast flow problems, Multiscale Model. Simul., 16, 3, 1227-1257 (2018) · Zbl 1407.65257
[110] Ohlberger, M., Error control based model reduction for multiscale problems, (Proceedings of the Conference ALGORITMY (2015)), 1-10, URL http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/310 · Zbl 1278.15008
[111] Henning, P.; Ohlberger, M., On the implementation of a heterogeneous multiscale finite element method for nonlinear elliptic problems, (Advances in DUNE. (2012), Springer: Springer Berlin), 143-155
[112] Milk, R.; Kaulmann, S., DUNE multiscale (2015)
[113] Bastian, P.; Engwer, C.; Fahlke, J.; Geveler, M.; Göddeke, D.; Iliev, O.; Ippisch, O.; Milk, R.; Mohring, J.; Müthing, S.; Ohlberger, M.; Ribbrock, D.; Turek, S., Advances concerning multiscale methods and uncertainty quantification in EXA-DUNE, (Software for Exascale Computing - SPPEXA 2013-2015. Software for Exascale Computing - SPPEXA 2013-2015, Lecture Notes in Computational Science and Engineering (2016), Springer Verlag), 25-43
[114] P. Bastian, M. Altenbernd, N. Dreier, C. Engwer, J. Fahlke, R. Fritze, M. Geveler, D. Göddeke, O. Iliev, O. Ippisch, J. Mohring, J. Müthing, M. Ohlberger, D. Ribbrock, N. Shegunov, S. Turek, EXA-DUNE — Flexible PDE Solvers, Numerical Methods and Applications, in: Bungartz, H.-J. and Nagel, W.E., Software for Exascale Computing - SPPEXA 2016-2018, Springer Lecture Notes in Computational Science and Engineering.
[115] Schindler, F.; Milk, R., DUNE generic discretization toolbox (2015)
[116] Milk, R.; Schindler, F.; Leibner, T., Extending DUNE: The dune-xt modules, Arch. Numer. Softw., 5, 1, 193-216 (2017)
[117] Engwer, C.; Fahlke, J., Scalable hybrid parallelization strategies for the DUNE grid interface, (Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2013. Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2013, Lecture Notes in Computational Science and Engineering, vol. 103 (2014)), 583-590 · Zbl 1320.65173
[118] Davis, T. A., Algorithm 832: UMFPACK v4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30, 2, 196-199 (2004) · Zbl 1072.65037
[119] Milk, R.; Mohring, J., DUNE-mlmc (SPPEXA AnPleMeet ’16) (2015)
[120] Fischer, P.; Min, M.; Rathnayake, T.; Dutta, S.; Kolev, T.; Dobrev, V.; Camier, J.-S.; Kronbichler, M.; Warburton, T.; Swirydowicz, K.; Brown, J., Scalability of high-performance PDE solvers (2020), arXiv:2004.06722
[121] Kronbichler, M.; Kormann, K., Fast matrix-free evaluation of discontinuous Galerkin finite element operators, ACM Trans. Math. Software, 45, 3 (2019) · Zbl 07193378
[122] Orszag, S. A., Spectral methods for problems in complex geometries, J. Comput. Phys., 37, 1, 70-92 (1980) · Zbl 0476.65078
[123] Müthing, S.; Piatkowski, M.; Bastian, P., High-performance implementation of matrix-free high-order discontinuous Galerkin methods, Int. J. High Perform. Comput. Appl. (2018), Accepted to arXiv:1711.10885
[124] Bastian, P.; Müller, E. H.; Müthing, S.; Piatkowski, M., Matrix-free multigrid block-preconditioners for higher order discontinuous Galerkin discretisations, J. Comput. Phys. (2019)
[125] Piatkowski, M.; Müthing, S.; Bastian, P., A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations, J. Comput. Phys., 356, 220-239 (2018) · Zbl 1380.76044
[126] Bosilca, G.; Bouteiller, A.; Guermouche, A.; Herault, T.; Robert, Y.; Sens, P.; Dongarra, J., Failure detection and propagation in HPC systems, (SC’16: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2016), IEEE), 312-322
[127] Bland, W.; Bouteiller, A.; Herault, T.; Hursey, J.; Bosilca, G.; Dongarra, J. J., An evaluation of user-level failure mitigation support in MPI, (European MPI Users’ Group Meeting (2012), Springer), 193-203
[128] Engwer, C.; Altenbernd, M.; Dreier, N.-A.; Göddeke, D., A high-level c++ approach to manage local errors, asynchrony and faults in an MPI application, (2018 26th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP) (2018), IEEE), 714-721, arXiv:1804.04481
[129] Ghysels, P.; Vanroose, W., Hiding global synchronization latency in the preconditioned conjugate gradient algorithm, Parallel Comput., 40, 7, 224-238 (2014)
[130] Müthing, S.; Ribbrock, D.; Göddeke, D., Integrating multi-threading and accelerators into DUNE-istl, (Proceedings of ENUMATH 2013, Vol. 103 (2014), Springer), 601-609 · Zbl 1320.65064
[131] Kreutzer, M.; Hager, G.; Wellein, G.; Fehske, H.; Bishop, A. R., A unified sparse matrix data format for modern processors with wide SIMD units, SIAM J. Sci. Comput., 36, 5, C401-C423 (2014) · Zbl 1307.65055
[132] Kempf, D.; Heß, R.; Müthing, S.; Bastian, P., Automatic code generation for high-performance discontinuous Galerkin methods on modern architectures (2018), ArXiv e-prints arXiv:1812.08075
[133] Kempf, D.; Bastian, P., An HPC perspective on generative programming, (Proceedings of the 14th International Workshop on Software Engineering for Science (2019), IEEE Press), 9-16
[134] Klöckner, A., Loo.py: Transformation-based code generation for GPUs and CPUs, (Proceedings of ACM SIGPLAN International Workshop on Libraries, Languages, and Compilers for Array Programming. Proceedings of ACM SIGPLAN International Workshop on Libraries, Languages, and Compilers for Array Programming, ARRAY’14 (2014), ACM: ACM New York, NY, USA), 82:82-82:87
[135] Bergen, B.; Gradl, T.; Hulsemann, F.; Rüde, U., A massively parallel multigrid method for finite elements, Comput. Sci. Eng., 8, 6, 56-62 (2006)
[136] MacDonald, A. E.; Middlecoff, J.; Henderson, T.; Lee, J.-L., A general method for modeling on irregular grids, Int. J. High Perform. Comput. Appl., 25, 4, 392-403 (2011)
[137] Bercea, G.; McRae, A. T.T.; Ham, D. A.; Mitchell, L.; Rathgeber, F.; Nardi, L.; Luporini, F.; Kelly, P. H.J., A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in firedrake, Geosci. Model Dev., 9, 10, 3803-3815 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.