FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF
We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady i...
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| Format: | Journal article |
| Language: | English |
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2011
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| author | Barrett, J Sueli, E |
| author_facet | Barrett, J Sueli, E |
| author_sort | Barrett, J |
| collection | OXFORD |
| description | We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ ℝ d, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function β L(̇):= min(̇,L) in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker-Planck-Navier-Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions. We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms, and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm. © EDP Sciences, SMAI, 2010. |
| first_indexed | 2024-03-06T22:06:08Z |
| format | Journal article |
| id | oxford-uuid:5033cf47-5ffe-44af-a052-bbe3bfa5b1c5 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:06:08Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:5033cf47-5ffe-44af-a052-bbe3bfa5b1c52022-03-26T16:12:09ZFINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFFJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5033cf47-5ffe-44af-a052-bbe3bfa5b1c5EnglishSymplectic Elements at Oxford2011Barrett, JSueli, EWe construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ ℝ d, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function β L(̇):= min(̇,L) in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker-Planck-Navier-Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions. We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms, and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm. © EDP Sciences, SMAI, 2010. |
| spellingShingle | Barrett, J Sueli, E FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title | FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title_full | FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title_fullStr | FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title_full_unstemmed | FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title_short | FINITE ELEMENT APPROXIMATION OF KINETIC DILUTE POLYMER MODELS WITH MICROSCOPIC CUT-OFF |
| title_sort | finite element approximation of kinetic dilute polymer models with microscopic cut off |
| work_keys_str_mv | AT barrettj finiteelementapproximationofkineticdilutepolymermodelswithmicroscopiccutoff AT suelie finiteelementapproximationofkineticdilutepolymermodelswithmicroscopiccutoff |