Optimal Constrained Interpolation in Mesh-Adaptive Finite Element Modeling
Abstract
Mesh-to-mesh Galerkin $L^2$ projection allows piecewise polynomial unstructured finite element data to be interpolated between two nonmatching unstructured meshes of the same domain. The interpolation is by definition optimal in an $L^2$ sense, and subject to fairly weak assumptions conserves the integral of an interpolated function. However other properties, such as the $L^2$ norm, or the weak divergence of a vector-valued function, can still be adversely affected by the interpolation. This is an important issue for calculations in which numerical dissipation should be minimized, or for simulations of incompressible flow. This paper considers extensions to mesh-to-mesh Galerkin $L^2$ projection which are $L^2$ optimal and ensure exact conservation of key discrete properties, including preservation of both the $L^2$ norm and the integral, and preservation of both the $L^2$ norm and weak incompressibility. The accuracy of the interpolants is studied. The utility of the interpolants is studied via adaptive mesh simulations of the two-dimensional lock-exchange problem, which are simulated using a combination of Fluidity and the FEniCS system.
MSC codes
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
A. Adcroft and D. Marshall, How slippery are piecewise-constant coastlines in numerical ocean models?, Tellus A, 50 (1998), pp. 95--108.
2.
M. S. Aln\aes, A. Logg, K.-A. Mardal, O. Skavhaug, and H. P. Langtangen, Unified framework for finite element assembly, Int. J. Comput. Sci. Engrg., 4 (2009), pp. 231--244.
3.
M. S. Aln\aes, A. Logg, K. B. Ølgaard, M. E. Rognes, and G. N. Wells, Unified form language: A domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Software, 40 (2014), 9.
4.
Applied Modelling and Computation Group (AMCG), Fluidity Manual, version 4.1.12, Department of Earth Science and Engineering, Imperial College London, London, UK, 2015.
5.
A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I, J. Comput. Phys., 1 (1966), pp. 119--143.
6.
D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), pp. 281--354.
7.
I. Babuška, The finite element method with Lagrangian multipliers, Numer. Math., 20 (1973), pp. 179--192.
8.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini, and H. Zhang, PETSc Users Manual, Tech. Report ANL-95/11, Revision 3.6, Argonne National Laboratory, Lemont, IL, 2015.
9.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, S. Zampini, and H. Zhang, PETSc Web Page, http://www.mcs.anl.gov/petsc. (Accessed 04-25-17.)
10.
S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object-oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser Boston, Inc., Boston, 1997, pp. 163--202.
11.
F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267--279.
12.
T. B. Benjamin, Gravity currents and related phenomena, J. Fluid Mech., 31 (1968), pp. 209--248.
13.
P.-E. Bernard, N. Chevaugeon, V. Legat, E. Deleersnijder, and J.-F. Remacle, High-order h-adaptive discontinuous Galerkin methods for ocean modelling, Ocean Dynamics, 57 (2007), pp. 109--121.
14.
S. Blaise and A. St-Cyr, A dynamic hp-adaptive discontinuous Galerkin method for shallow-water flows on the sphere with application to a global tsunami simulation, Monthly Weather Rev., 140 (2012), pp. 978--996.
15.
P. Bochev and M. Shashkov, Constrained interpolation (remap) of divergence-free fields, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 511--530.
16.
H. Borouchaki, F. Hecht, and P. J. Frey, Mesh gradation control, Int. J. Numer. Methods Engrg., 43 (1998), pp. 1143--1165, https://doi.org/10.1002/(SICI)1097-0207(19981130)43:6<1143::AID-NME470>3.0.CO;2-I,
17.
F. Botelho, Functional Analysis and Applied Optimization in Banach Spaces: Applications to Non-convex Variational Models, Springer, New York, 2014.
18.
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge, 8 (1974), pp. 129--151.
19.
A. G. Buchan, P. E. Farrell, G. J. Gorman, A. J. H. Goddard, M. D. Eaton, E. T. Nygaard, P. L. Angelo, R. P. Smedley-Stevenson, S. R. Merton, and P. N. Smith, The immersed body supermeshing method for modelling reactor-physics problems with complex internal structures, Ann. Nuclear Energy, 63 (2014), pp. 399--408.
20.
G. F. Carey, G. Bicken, V. Carey, C. Berger, and J. Sanchez, Locally constrained projections on grids, Int. J. Numer. Methods Engrg., 50 (2001), pp. 549--577.
21.
L. Chen, P. Sun, and J. Xu, Optimal anisotropic meshes for minimizing interpolation errors in ${L}^p$-norm, Math. Comp., 76 (2007), pp. 179--204, https://doi.org/10.1090/S0025-5718-06-01896-5.
22.
T. J. Chung, Finite Element Analysis in Fluid Dynamics, McGraw-Hill, New York, 1978.
23.
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Appl. Math. 40, SIAM, Philadelphia, 2002, https://doi.org/10.1137/1.9780898719208.
24.
C. J. Cotter and D. A. Ham, Numerical wave propagation for the triangular $P1_{DG}$--$P2$ finite element pair, J. Comput. Phys., 230 (2011), pp. 2806--2820.
25.
C. J. Cotter, D. A. Ham, and C. C. Pain, A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling, Ocean Model., 26 (2009), pp. 86--90.
26.
C. J. Cotter, D. A. Ham, C. C. Pain, and S. Reich, LBB stability of a mixed Galerkin finite element pair for fluid flow simulations, J. Comput. Phys., 228 (2009), pp. 336--348.
27.
C. J. Cotter and J. Shipton, Mixed finite elements for numerical weather prediction, J. Comput. Phys., 231 (2012), pp. 7076--7091.
28.
C. J. Cotter and J. Thuburn, A finite element exterior calculus framework for the rotating shallow-water equations, J. Comput. Phys., 257 (2014), pp. 1506--1526.
29.
S. Danilov, Ocean modeling on unstructured meshes, Ocean Model., 69 (2013), pp. 195--210.
30.
D. R. Davies, C. R. Wilson, and S. C. Kramer, Fluidity: A fully unstructured anisotropic adaptive mesh computational modeling framework for geodynamics, Geochem. Geophys. Geosyst., 12 (2011), Q06001.
31.
D. H. Eberly, $3$D Game Engine Design: A Practical Approach to Real-Time Computer Graphics, 2nd ed., CRC Press, Boca Raton, FL, 2007.
32.
C. Eskilsson, An hp-adaptive discontinuous Galerkin method for shallow water flows, Int. J. Numer. Methods Fluids, 67 (2011), pp. 1605--1623.
33.
R. D. Falgout and U. M. Yang, hypre: A library of high performance preconditioners, in Proceedings of Computational Science -- ICCS 2002, P. M. A. Sloot et al., eds., Lecture Notes in Comput. Sci. 2331, Springer, Berlin, 2002, pp. 632--641.
34.
P. E. Farrell, Galerkin Projection of Discrete Fields via Supermesh Construction, Ph.D. thesis, Imperial College London, 2009.
35.
P. E. Farrell, The addition of fields on different meshes, J. Comput. Phys., 230 (2011), pp. 3265--3269.
36.
P. E. Farrell and J. R. Maddison, Conservative interpolation between volume meshes by local Galerkin projection, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 89--100.
37.
P. E. Farrell, M. D. Piggott, C. C. Pain, G. J. Gorman, and C. R. Wilson, Conservative interpolation between unstructured meshes via supermesh construction, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632--2642.
38.
G. J. Fix, Finite element models for ocean circulation problems, SIAM J. Appl. Math., 29 (1975), pp. 371--387, https://doi.org/10.1137/0129031.
39.
R. Ford, C. C. Pain, M. D. Piggott, A. J. H. Goddard, C. R. E. de Oliveira, and A. P. Umpleby, A nonhydrostatic finite-element model for three-dimensional stratified oceanic flows. Part I: Model formulation, Monthly Weather Rev., 132 (2004), pp. 2816--2831.
40.
M. J. Gander and C. Japhet, An algorithm for non-matching grid projections with linear complexity, in Domain Decomposition Methods in Science and Engineering XVIII, M. Bercovier et al., eds., Lecture Notes in Comput. Sci. Eng. 70, Springer, Berlin, 2009, pp. 185--192.
41.
M. J. Gander and C. Japhet, Algorithm 932: PANG: Software for nonmatching grid projections in 2D and 3D with linear complexity, ACM Trans. Math. Software, 40 (2013), 6.
42.
P. L. George and H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements, Hermes, Paris, 1998.
43.
C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Engrg., 79 (2009), pp. 1309--1331.
44.
G. J. Gorman, C. C. Pain, M. D. Piggott, A. P. Umpleby, P. E. Farrell, and J. R. Maddison, Interleaved parallel tetrahedral mesh optimisation and dynamic load-balancing, in Proceedings of the International Conference on Adaptive Modeling and Simulation (ADMOS 2009), Brussells, Belgium, P. Bouillard and P. Díez, eds., 2009, pp. 101--104.
45.
J. Grandy, Conservative remapping and region overlays by intersecting arbitrary polyhedra, J. Comput. Phys., 148 (1999), pp. 433--466.
46.
P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method. Vol. 1: Advection-Diffusion, John Wiley & Sons, New York, 2000.
47.
P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method. Vol. 2: Isothermal Laminar Flow, John Wiley & Sons, New York, 2000.
48.
S. M. Griffies, A. J. Adcroft, H. Banks, C. W. Böning, E. P. Chassignet, G. Danabasoglu, S. Danilov, E. Deleersnijder, H. Drange, M. England, B. Fox-Kemper, R. Gerdes, A. Gnanadesikan, R. J. Greatbatch, R. W. Hallberg, E. Hanert, M. J. Harrison, S. Legg, C. M. Little, G. Madec, S. J. Marsland, M. Nikurashin, A. Pirani, H. L. Simmons, J. Schröter, B. L. Samuels, A.-M. Treguier, J. R. Toggweiler, H. Tsujino, G. K. Vallis, and L. White, Problems and prospects in large-scale ocean circulation models, in Proceedings of OceanObs'09: Sustained Ocean Observations and Information for Society, (Venice, Italy), vol. 2, J. Hall, D. E. Harrison, and D. Stammer, eds., ESA Publication WPP-306, 2010.
49.
S. M. Griffies, C. Böning, F. O. Bryan, E. P. Chassignet, R. Gerdes, H. Hasumi, A. Hirst, A. M. Treguier, and D. Webb, Developments in ocean climate modelling, Ocean Model., 2 (2000), pp. 123--192.
50.
J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 6011--6045.
51.
C. Härtel, E. Meiburg, and F. Necker, Analysis and direct numerical simulation of the flow at a gravity current head. Part $1$. Flow topology and front speed for slip and no-slip boundaries, J. Fluid Mech., 418 (2000), pp. 189--212, https://doi.org/10.1017/S0022112000001221.
52.
V. E. Henson and U. M. Yang, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155--177.
53.
H. R. Hiester, The Application of Adaptive Mesh Techniques to Numerical Simulations of Gravity Current Flows, PhD thesis, Imperial College London, 2011.
54.
H. R. Hiester, M. D. Piggott, and P. A. Allison, The impact of mesh adaptivity on the gravity current front speed in a two-dimensional lock-exchange, Ocean Model., 38 (2011), pp. 1--21.
55.
H. R. Hiester, M. D. Piggott, P. E. Farrell, and P. A. Allison, Assessment of spurious mixing in adaptive mesh simulations of the two-dimensional lock-exchange, Ocean Model., 73 (2014), pp. 30--44.
56.
R. C. Kirby, Algorithm 839: FIAT, a new paradigm for computing finite element basis functions, ACM Trans. Math. Software, 30 (2004), pp. 502--516.
57.
R. C. Kirby and A. Logg, A compiler for variational forms, ACM Trans. Math. Software, 32 (2006), pp. 417--444.
58.
S. C. Kramer, C. J. Cotter, and C. C. Pain, Solving the Poisson equation on small aspect ratio domains using unstructured meshes, Ocean Model., 35 (2010), pp. 253--263.
59.
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, second English edition, revised and enlarged, Math. Appl. 2, translated from the Russian by R. A. Silverman and J. Chu, Gordon and Breach Science Publishers, New York, 1969.
60.
R. J. LeVeque, Finite--Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.
61.
A. Logg, K.-A. Mardal, and G. N. Wells, eds., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lecture Notes in Comput. Sci. Engrg. 84, Springer, Berlin, 2012.
62.
A. Logg and G. N. Wells, DOLFIN: Automated finite element computing, ACM Trans. Math. Software, 37 (2010), 20.
63.
A. Loseille and F. Alauzet, Continuous mesh framework Part II: Validations and applications, SIAM J. Numer. Anal., 49 (2011), pp. 61--86, https://doi.org/10.1137/10078654X.
64.
A. E. MacDonald, J. Middlecoff, T. Henderson, and J.-L. Lee, A general method for modeling on irregular grids, Int. J. High Performance Comput. Appl., 25 (2011), pp. 392--403.
65.
J. R. Maddison, C. J. Cotter, and P. E. Farrell, Geostrophic balance preserving interpolation in mesh adaptive linearised shallow-water ocean modelling, Ocean Model., 37 (2011), pp. 35--48.
66.
J. R. Maddison and P. E. Farrell, Directional integration on unstructured meshes via supermesh construction, J. Comput. Phys., 231 (2012), pp. 4422--4432.
67.
G. R. Markall, F. Rathgeber, L. Mitchell, N. Loriant, C. Bertolli, D. A. Ham, and P. H. J. Kelly, Performance-portable finite element assembly using PyOP$2$ and FEniCS, in Proceedings of Supercomputing: 28th International Supercomputing Conference (ISC 2013, Leipzig, Germany), J. M. Kunkel, T. Ludwig, and H. W. Meuer, eds., Lecture Notes in Comput. Sci. 7905, Springer, Berlin, 2013, pp. 279--289.
68.
D. R. Munday, D. P. Marshall, and M. D. Piggott, Idealised flow past an island in a dynamically adaptive finite element model, Ocean Dynamics, 60 (2010), pp. 835--850.
69.
K. B. Ølgaard and G. N. Wells, Optimizations for quadrature representations of finite element tensors through automated code generation, ACM Trans. Math. Software, 37 (2010), 8.
70.
C. C. Pain, M. D. Piggott, A. J. H. Goddard, F. Fang, G. J. Gorman, D. P. Marshall, M. D. Eaton, P. W. Power, and C. R. E. de Oliveira, Three-dimensional unstructured mesh ocean modelling, Ocean Model., 10 (2005), pp. 5--33.
71.
C. C. Pain, A. P. Umpleby, C. R. E. De Oliveira, and A. J. H. Goddard, Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 3771--3796.
72.
M. D. Piggott, P. E. Farrell, C. R. Wilson, G. J. Gorman, and C. C. Pain, Anisotropic mesh adaptivity for multi-scale ocean modelling, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), pp. 4591--4611.
73.
M. D. Piggott, G. J. Gorman, C. C. Pain, P. A. Allison, A. S. Candy, B. T. Martin, and M. R. Wells, A new computational framework for multi-scale ocean modelling based on adapting unstructured meshes, Int. J. Numer. Methods Fluids, 56 (2008), pp. 1003--1015.
74.
M. D. Piggott, C. C. Pain, G. J. Gorman, D. P. Marshall, and P. D. Killworth, Unstructured adaptive meshes for ocean modeling, in Ocean Modeling in an Eddying Regime, M. W. Hecht and H. Hasumi, eds., Geophys. Monograph Ser. 177, American Geophysical Union, Washington, DC, 2008, pp. 383--408.
75.
M. D. Piggott, C. C. Pain, G. J. Gorman, P. W. Power, and A. J. H. Goddard, h, r, and hr adaptivity with applications in numerical ocean modelling, Ocean Model., 10 (2005), pp. 95--113.
76.
T. Ringler, M. Petersen, R. L. Higdon, D. Jacobsen, P. W. Jones, and M. Maltrud, A multi-resolution approach to global ocean modeling, Ocean Model., 69 (2013), pp. 211--232.
77.
T. D. Ringler and D. A. Randall, A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid, Monthly Weather Rev., 130 (2002), pp. 1397--1410.
78.
R. Salmon, Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations, J. Atmospheric Sci., 61 (2004), pp. 2016--2036.
79.
J. E. Simpson, Gravity Currents in the Environment and the Laboratory, Ellis Horwood Limited, Hempstead, UK, 1987.
80.
J. Southern, G. J. Gorman, M. D. Piggott, and P. E. Farrell, Parallel anisotropic mesh adaptivity with dynamic load balancing for cardiac electrophysiology, J. Comput. Sci., 3 (2012), pp. 8--16.
81.
A. H. Stroud, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, NJ, 1971.
82.
P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), pp. 995--1011.
83.
Y.-H. Tseng and J. H. Ferziger, Mixing and available potential energy in stratified flows, Phys. Fluids, 13 (2001), pp. 1281--1293, https://doi.org/10.1063/1.1358307.
84.
P. Vaněk, J. Mandel, and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179--196.
85.
Y. V. Vasilevskii and K. N. Lipnikov, An adaptive algorithm for quasioptimal mesh generation, Comput. Math. Math. Phys., 39 (1999), pp. 1468--1486.
86.
A. Viré, J. Xiang, F. Milthaler, P. E. Farrell, M. D. Piggott, J.-P. Latham, D. Pavlidis, and C. C. Pain, Modelling of fluid-solid interactions using an adaptive mesh fluid model coupled with a combined finite-discrete element model, Ocean Dynamics, 62 (2012), pp. 1487--1501.
87.
Q. Wang, H. Zhou, and W. Decheng, Numerical simulation of wind turbine blade-tower interaction, J. Marine Sci. Appl., 11 (2012), pp. 321--327.
88.
L. White, V. Legat, and E. Deleersnijder, Tracer conservation for three-dimensional, finite-element, free-surface ocean modeling on moving prismatic meshes, Monthly Weather Rev., 136 (2008), pp. 420--442.
89.
C. Wilson, Modelling Multiple-Material Flows on Adaptive Unstructured Meshes, Ph.D. thesis, Imperial College London, 2009.
90.
K. B. Winters and E. A. D'Asaro, Diascalar flux and the rate of fluid mixing, J. Fluid Mech., 317 (1996), pp. 179--193, https://doi.org/10.1017/S0022112096000717.
91.
K. B. Winters, P. N. Lombard, J. J. Riley, and E. A. D'Asaro, Available potential energy and mixing in density-stratified fluids, J. Fluid Mech., 289 (1995), pp. 115--128, https://doi.org/10.1017/S002211209500125X.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A2257 - A2286
DOI: 10.1137/15M102054X
ISSN (online): 1095-7197
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 7 May 2015
Accepted: 2 December 2016
Published online: 28 September 2017
MSC codes
MSC codes
Authors
Funding Information
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/G036136/1
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.