2016, 9(2): 475-500. doi: 10.3934/dcdss.2016008

On the space separated representation when addressing the solution of PDE in complex domains

1. 

Notre Dame University-Louaize, Zouk Mosbeh P.O. Box 72, Lebanon

2. 

GeM Institute, UMR CNRS - Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France, France

3. 

LHEEA, UMR CNRS - Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France

4. 

High Performance Computing Institute - Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101, F-44321 Nantes cedex 3, France

5. 

P' Institute, UPR CNRS - University of Poitiers & ENSMA, 11 Boulevard Marie et Pierre Curie, BP 30179, F-86962 Futuroscope Chasseneuil cedex, France

Received  September 2014 Revised  October 2015 Published  March 2016

Separated representations allow impressive computational CPU time savings when applied in different fields of computational mechanics. They have been extensively used for solving models defined in multidimensional spaces coming from (i) its proper physics, (ii) model parameters that were introduced as extra-coordinates and (iii) 3D models when the solution can be separated as a finite sum of functional products involving lower dimensional spaces. The last route is especially suitable when models are defined in hexahedral domains. When it is not the case, different possibilities exist and were considered in our former works. In the present work, we are analyzing two alternative routes. The first one consists of immersing the real non-separable domain into a fully separable hexahedral domain. The second procedure consists in applying a geometrical transformation able to transform the real domain into a hexahedra in which the model is solved by using a fully separated representation of the unknown field.
Citation: Chady Ghnatios, Guangtao Xu, Adrien Leygue, Michel Visonneau, Francisco Chinesta, Alain Cimetiere. On the space separated representation when addressing the solution of PDE in complex domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 475-500. doi: 10.3934/dcdss.2016008
References:
[1]

A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids,, Journal of Non-Newtonian Fluid Mechanics, 139 (2006), 153. doi: 10.1016/j.jnnfm.2006.07.007.

[2]

A. Ammar, D. Ryckelynck, F. Chinesta and R. Keunings, On the reduction of kinetic theory models related to finitely extensible dumbbells,, Journal of Non-Newtonian Fluid Mechanics, 134 (2006), 136. doi: 10.1016/j.jnnfm.2006.01.007.

[3]

A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: Transient simulation using space-time separated representation,, Journal of Non-Newtonian Fluid Mechanics, 144 (2007), 98.

[4]

A. Ammar, F. Chinesta and P. Joyot, The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions,, International Journal for Multiscale Computational Engineering, 6/3 (2008), 191. doi: 10.1615/IntJMultCompEng.v6.i3.20.

[5]

A. Ammar, M. Normandin and F. Chinesta, Solving parametric complex fluids models in rheometric flows,, Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 1588. doi: 10.1016/j.jnnfm.2010.08.006.

[6]

A. Ammar, E. Cueto and F. Chinesta, Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions,, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 960. doi: 10.1002/cnm.2476.

[7]

A. Ammar, A. Huerta, F. Chinesta, E. Cueto and A. Leygue, Parametric solutions involving geometry: A step towards efficient shape optimization,, Computer Methods in Applied Mechanics and Engineering, 268 (2014), 178. doi: 10.1016/j.cma.2013.09.003.

[8]

N. Bellomo, Modeling Complex Living Systems,, Birkhauser, (2008).

[9]

R. A. Bialecki, A. J. Kassab and A. Fic, Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis,, Int. J. Numer. Meth. Engrg., 62 (2005), 774.

[10]

B. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, in: Kinetic Theory, Vol 2,, John Wiley & Sons, (1987).

[11]

B. Bognet, A. Leygue, F. Chinesta, A. Poitou and F. Bordeu, Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity,, Computer Methods in Applied Mechanics and Engineering, 201 (2012), 1. doi: 10.1016/j.cma.2011.08.025.

[12]

B. Bognet, A. Leygue and F. Chinesta, Separated representations of 3D elastic solutions in shell geometries,, Advanced Modelling and Simulation in Engineering Sciences, 1 (2014). doi: 10.1186/2213-7467-1-4.

[13]

A. Bruno-Alfonso, L. Cabezas-Gomez and H. Aparecido-Navarro, Alternate treatments of jacobian singularities in polar coordinates within finite-difference schemes,, World Journal of Modelling and Simulation, 8 (2012), 163.

[14]

J. Burkardt, M. Gunzburger and H.-Ch. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows,, Comput. Methods Appl. Mech. Engrg., 196 (2006), 337. doi: 10.1016/j.cma.2006.04.004.

[15]

E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational Quantum Chemistry: A primer, in Handbook of Numerical Analysis, Vol X,, Elsevier, (2003).

[16]

F. Chinesta, A. Ammar, A. Leygue and R. Keunings, An overview of the Proper Generalized Decomposition with applications in computational rheology,, Journal of Non Newtonian Fluid Mechanics, 166 (2011), 578. doi: 10.1016/j.jnnfm.2010.12.012.

[17]

F. Chinesta, P. Ladeveze and E. Cueto, A short review in model order reduction based on Proper Generalized Decomposition,, Archives of Computational Methods in Engineering, 18 (2011), 395. doi: 10.1007/s11831-011-9064-7.

[18]

F. Chinesta, A. Leygue, F. Bordeu, J. V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar and A. Huerta, Parametric PGD based computational vademecum for efficient design, optimization and control,, Archives of Computational Methods in Engineering, 20 (2013), 31. doi: 10.1007/s11831-013-9080-x.

[19]

F. Chinesta, A. Leygue, B. Bognet, Ch. Ghnatios, F. Poulhaon, F. Bordeu, A. Barasinski, A. Poitou, S. Chatel and S. Maison-Le-Poec, First steps towards an advanced simulation of composites manufacturing by automated tape placement,, International Journal of Material Forming, 7 (2014), 81. doi: 10.1007/s12289-012-1112-9.

[20]

F. Chinesta, R. Keunings and A. Leygue, The Proper Generalized Decomposition for advanced numerical simulations. A primer,, Springerbriefs, (2014). doi: 10.1007/978-3-319-02865-1.

[21]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313.

[22]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, An inversion method for harmonic functions reconstruction,, International Journal of Thermal Sciences, 41 (2002), 509. doi: 10.1016/S1290-0729(02)01344-3.

[23]

Ch. Ghnatios, F. Chinesta, E. Cueto, A. Leygue, P. Breitkopf and P. Villon, Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: Application to pultrusion,, Composites Part A, 42 (2011), 1169. doi: 10.1016/j.compositesa.2011.05.001.

[24]

Ch. Ghnatios, F. Masson, A. Huerta, E. Cueto, A. Leygue and F. Chinesta, Proper Generalized Decomposition based dynamic data-driven control of thermal processes,, Computer Methods in Applied Mechanics and Engineering, 213 (2012), 29. doi: 10.1016/j.cma.2011.11.018.

[25]

Ch. Ghnatios, F. Chinesta et Ch. Binetruy, The squeeze flow of composite laminates,, International Journal of Material Forming, ().

[26]

D. Gonzalez, A. Ammar, F. Chinesta and E. Cueto, Recent advances on the use of separated representations,, International Journal for Numerical Methods in Engineering, 81 (2010), 637. doi: 10.1002/nme.2710.

[27]

D. Gonzalez, F. Masson, F. Poulhaon, A. Leygue, E. Cueto and F. Chinesta, Proper Generalized Decomposition based dynamic data-driven inverse identification,, Mathematics and Computers in Simulation, 82 (2012), 1677. doi: 10.1016/j.matcom.2012.04.001.

[28]

M. D. Gunzburger, J. S. Peterson and J. N. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1030. doi: 10.1016/j.cma.2006.08.004.

[29]

P. Ladeveze, The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables,, Comptes Rendus Académie des Sciences Paris, 309 (1989), 1095.

[30]

P. Ladevèze, J.-C. Passieux and D. Néron, The latin multiscale computational method and the proper generalized decomposition,, Computer Methods In Applied Mechanics and Engineering, 199 (2010), 1287. doi: 10.1016/j.cma.2009.06.023.

[31]

H. Lamari, A. Ammar, A. Leygue and F. Chinesta, On the solution of the multidimensional Langer's equation by using the Proper Generalized Decomposition Method for modeling phase transitions,, Modelling and Simulation in Materials Science and Engineering, 20 (2012). doi: 10.1088/0965-0393/20/1/015007.

[32]

A. Leygue, F. Chinesta, M. Beringhier, T. L. Nguyen, J. C. Grandidier, F. Pasavento and B. Schrefler, Towards a framework for non-linear thermal models in shell domains,, International Journal of Numerical Methods for Heat and Fluid Flow, 23 (2013), 55. doi: 10.1108/09615531311289105.

[33]

Y. Maday and E. M. Ronquist, A reduced-basis element method,, C. R. Acad. Sci. Paris, 335 (2002), 195. doi: 10.1016/S1631-073X(02)02427-5.

[34]

Y. Maday, A. T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations,, Journal of Scientific Computing, 17 (2002), 437. doi: 10.1023/A:1015145924517.

[35]

Y. Maday and E. M. Ronquist, The reduced basis element method: Application to a thermal fin problem,, SIAM J. Sci. Comput., 26 (2004), 240. doi: 10.1137/S1064827502419932.

[36]

S. Niroomandi, I. Alfaro, E. Cueto and F. Chinesta, Accounting for large deformations in real-time simulations of soft tissues based on reduced order models,, Computer Methods and Programs in Biomedicine, 105 (2012), 1.

[37]

A. Nouy, Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems,, Archives of Computational Methods in Engineering - State of the Art Reviews, 17 (2010), 403. doi: 10.1007/s11831-010-9054-1.

[38]

H. M. Park and D. H. Cho, The use of the Karhunen-Loève decomposition for the modelling of distributed parameter systems,, Chem. Engineer. Science, 51 (1996), 81.

[39]

E. Pruliere, F. Chinesta and A. Ammar, On the deterministic solution of multidimensional parametric models by using the Proper Generalized Decomposition,, Mathematics and Computers in Simulation, 81 (2010), 791. doi: 10.1016/j.matcom.2010.07.015.

[40]

G. Rozza, D. B. P. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - application to transport and continuum mechanics,, Archives of Computational Methods in Engineering, 15 (2008), 229. doi: 10.1007/s11831-008-9019-9.

[41]

D. Ryckelynck, L. Hermanns, F. Chinesta and E. Alarcon, An efficient a priori model reduction for boundary element models,, Engineering Analysis with Boundary Elements, 29 (2005), 796. doi: 10.1016/j.enganabound.2005.04.003.

[42]

D. Ryckelynck, F. Chinesta, E. Cueto and A. Ammar, On the a priori model reduction: Overview and recent developments,, Archives of Computational Methods in Engineering, 13 (2006), 91. doi: 10.1007/BF02905932.

[43]

F. Schmidt, N. Pirc, M. Mongeau and F. Chinesta, Efficient mould cooling optimization by using model reduction,, International Journal of Material Forming, 4 (2011), 71.

[44]

K. Veroy and A. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds,, Int. J. Numer. Meth. Fluids, 47 (2005), 773. doi: 10.1002/fld.867.

show all references

References:
[1]

A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids,, Journal of Non-Newtonian Fluid Mechanics, 139 (2006), 153. doi: 10.1016/j.jnnfm.2006.07.007.

[2]

A. Ammar, D. Ryckelynck, F. Chinesta and R. Keunings, On the reduction of kinetic theory models related to finitely extensible dumbbells,, Journal of Non-Newtonian Fluid Mechanics, 134 (2006), 136. doi: 10.1016/j.jnnfm.2006.01.007.

[3]

A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: Transient simulation using space-time separated representation,, Journal of Non-Newtonian Fluid Mechanics, 144 (2007), 98.

[4]

A. Ammar, F. Chinesta and P. Joyot, The nanometric and micrometric scales of the structure and mechanics of materials revisited: An introduction to the challenges of fully deterministic numerical descriptions,, International Journal for Multiscale Computational Engineering, 6/3 (2008), 191. doi: 10.1615/IntJMultCompEng.v6.i3.20.

[5]

A. Ammar, M. Normandin and F. Chinesta, Solving parametric complex fluids models in rheometric flows,, Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 1588. doi: 10.1016/j.jnnfm.2010.08.006.

[6]

A. Ammar, E. Cueto and F. Chinesta, Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions,, International Journal for Numerical Methods in Biomedical Engineering, 28 (2012), 960. doi: 10.1002/cnm.2476.

[7]

A. Ammar, A. Huerta, F. Chinesta, E. Cueto and A. Leygue, Parametric solutions involving geometry: A step towards efficient shape optimization,, Computer Methods in Applied Mechanics and Engineering, 268 (2014), 178. doi: 10.1016/j.cma.2013.09.003.

[8]

N. Bellomo, Modeling Complex Living Systems,, Birkhauser, (2008).

[9]

R. A. Bialecki, A. J. Kassab and A. Fic, Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis,, Int. J. Numer. Meth. Engrg., 62 (2005), 774.

[10]

B. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, Dynamics of polymeric liquids, in: Kinetic Theory, Vol 2,, John Wiley & Sons, (1987).

[11]

B. Bognet, A. Leygue, F. Chinesta, A. Poitou and F. Bordeu, Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity,, Computer Methods in Applied Mechanics and Engineering, 201 (2012), 1. doi: 10.1016/j.cma.2011.08.025.

[12]

B. Bognet, A. Leygue and F. Chinesta, Separated representations of 3D elastic solutions in shell geometries,, Advanced Modelling and Simulation in Engineering Sciences, 1 (2014). doi: 10.1186/2213-7467-1-4.

[13]

A. Bruno-Alfonso, L. Cabezas-Gomez and H. Aparecido-Navarro, Alternate treatments of jacobian singularities in polar coordinates within finite-difference schemes,, World Journal of Modelling and Simulation, 8 (2012), 163.

[14]

J. Burkardt, M. Gunzburger and H.-Ch. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows,, Comput. Methods Appl. Mech. Engrg., 196 (2006), 337. doi: 10.1016/j.cma.2006.04.004.

[15]

E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational Quantum Chemistry: A primer, in Handbook of Numerical Analysis, Vol X,, Elsevier, (2003).

[16]

F. Chinesta, A. Ammar, A. Leygue and R. Keunings, An overview of the Proper Generalized Decomposition with applications in computational rheology,, Journal of Non Newtonian Fluid Mechanics, 166 (2011), 578. doi: 10.1016/j.jnnfm.2010.12.012.

[17]

F. Chinesta, P. Ladeveze and E. Cueto, A short review in model order reduction based on Proper Generalized Decomposition,, Archives of Computational Methods in Engineering, 18 (2011), 395. doi: 10.1007/s11831-011-9064-7.

[18]

F. Chinesta, A. Leygue, F. Bordeu, J. V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar and A. Huerta, Parametric PGD based computational vademecum for efficient design, optimization and control,, Archives of Computational Methods in Engineering, 20 (2013), 31. doi: 10.1007/s11831-013-9080-x.

[19]

F. Chinesta, A. Leygue, B. Bognet, Ch. Ghnatios, F. Poulhaon, F. Bordeu, A. Barasinski, A. Poitou, S. Chatel and S. Maison-Le-Poec, First steps towards an advanced simulation of composites manufacturing by automated tape placement,, International Journal of Material Forming, 7 (2014), 81. doi: 10.1007/s12289-012-1112-9.

[20]

F. Chinesta, R. Keunings and A. Leygue, The Proper Generalized Decomposition for advanced numerical simulations. A primer,, Springerbriefs, (2014). doi: 10.1007/978-3-319-02865-1.

[21]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, Solution of the Cauchy problem using iterated Tikhonov regularization,, Inverse Problems, 17 (2001), 553. doi: 10.1088/0266-5611/17/3/313.

[22]

A. Cimetiere, F. Delvare, M. Jaoua and F. Pons, An inversion method for harmonic functions reconstruction,, International Journal of Thermal Sciences, 41 (2002), 509. doi: 10.1016/S1290-0729(02)01344-3.

[23]

Ch. Ghnatios, F. Chinesta, E. Cueto, A. Leygue, P. Breitkopf and P. Villon, Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: Application to pultrusion,, Composites Part A, 42 (2011), 1169. doi: 10.1016/j.compositesa.2011.05.001.

[24]

Ch. Ghnatios, F. Masson, A. Huerta, E. Cueto, A. Leygue and F. Chinesta, Proper Generalized Decomposition based dynamic data-driven control of thermal processes,, Computer Methods in Applied Mechanics and Engineering, 213 (2012), 29. doi: 10.1016/j.cma.2011.11.018.

[25]

Ch. Ghnatios, F. Chinesta et Ch. Binetruy, The squeeze flow of composite laminates,, International Journal of Material Forming, ().

[26]

D. Gonzalez, A. Ammar, F. Chinesta and E. Cueto, Recent advances on the use of separated representations,, International Journal for Numerical Methods in Engineering, 81 (2010), 637. doi: 10.1002/nme.2710.

[27]

D. Gonzalez, F. Masson, F. Poulhaon, A. Leygue, E. Cueto and F. Chinesta, Proper Generalized Decomposition based dynamic data-driven inverse identification,, Mathematics and Computers in Simulation, 82 (2012), 1677. doi: 10.1016/j.matcom.2012.04.001.

[28]

M. D. Gunzburger, J. S. Peterson and J. N. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1030. doi: 10.1016/j.cma.2006.08.004.

[29]

P. Ladeveze, The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables,, Comptes Rendus Académie des Sciences Paris, 309 (1989), 1095.

[30]

P. Ladevèze, J.-C. Passieux and D. Néron, The latin multiscale computational method and the proper generalized decomposition,, Computer Methods In Applied Mechanics and Engineering, 199 (2010), 1287. doi: 10.1016/j.cma.2009.06.023.

[31]

H. Lamari, A. Ammar, A. Leygue and F. Chinesta, On the solution of the multidimensional Langer's equation by using the Proper Generalized Decomposition Method for modeling phase transitions,, Modelling and Simulation in Materials Science and Engineering, 20 (2012). doi: 10.1088/0965-0393/20/1/015007.

[32]

A. Leygue, F. Chinesta, M. Beringhier, T. L. Nguyen, J. C. Grandidier, F. Pasavento and B. Schrefler, Towards a framework for non-linear thermal models in shell domains,, International Journal of Numerical Methods for Heat and Fluid Flow, 23 (2013), 55. doi: 10.1108/09615531311289105.

[33]

Y. Maday and E. M. Ronquist, A reduced-basis element method,, C. R. Acad. Sci. Paris, 335 (2002), 195. doi: 10.1016/S1631-073X(02)02427-5.

[34]

Y. Maday, A. T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations,, Journal of Scientific Computing, 17 (2002), 437. doi: 10.1023/A:1015145924517.

[35]

Y. Maday and E. M. Ronquist, The reduced basis element method: Application to a thermal fin problem,, SIAM J. Sci. Comput., 26 (2004), 240. doi: 10.1137/S1064827502419932.

[36]

S. Niroomandi, I. Alfaro, E. Cueto and F. Chinesta, Accounting for large deformations in real-time simulations of soft tissues based on reduced order models,, Computer Methods and Programs in Biomedicine, 105 (2012), 1.

[37]

A. Nouy, Proper Generalized Decompositions and separated representations for the numerical solution of high dimensional stochastic problems,, Archives of Computational Methods in Engineering - State of the Art Reviews, 17 (2010), 403. doi: 10.1007/s11831-010-9054-1.

[38]

H. M. Park and D. H. Cho, The use of the Karhunen-Loève decomposition for the modelling of distributed parameter systems,, Chem. Engineer. Science, 51 (1996), 81.

[39]

E. Pruliere, F. Chinesta and A. Ammar, On the deterministic solution of multidimensional parametric models by using the Proper Generalized Decomposition,, Mathematics and Computers in Simulation, 81 (2010), 791. doi: 10.1016/j.matcom.2010.07.015.

[40]

G. Rozza, D. B. P. Huynh and A. T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - application to transport and continuum mechanics,, Archives of Computational Methods in Engineering, 15 (2008), 229. doi: 10.1007/s11831-008-9019-9.

[41]

D. Ryckelynck, L. Hermanns, F. Chinesta and E. Alarcon, An efficient a priori model reduction for boundary element models,, Engineering Analysis with Boundary Elements, 29 (2005), 796. doi: 10.1016/j.enganabound.2005.04.003.

[42]

D. Ryckelynck, F. Chinesta, E. Cueto and A. Ammar, On the a priori model reduction: Overview and recent developments,, Archives of Computational Methods in Engineering, 13 (2006), 91. doi: 10.1007/BF02905932.

[43]

F. Schmidt, N. Pirc, M. Mongeau and F. Chinesta, Efficient mould cooling optimization by using model reduction,, International Journal of Material Forming, 4 (2011), 71.

[44]

K. Veroy and A. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds,, Int. J. Numer. Meth. Fluids, 47 (2005), 773. doi: 10.1002/fld.867.

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