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Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2008 (English)In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 57, no 9, 1397-1420 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we develop an adaptive finite element method for heat transfer in incompressible fluid flow. The adaptive method is based on an a posteriori error estimate for the coupled problem, which identifies how accurately the flow and heat transfer problems must be solved in order to achieve overall accuracy in a specified goal quantity. The a posteriori error estimate is derived using duality techniques and is of dual weighted residual type. We consider, in particular, an a posteriori error estimate for a variational approximation of the integrated heat flux through the boundary of a hot object immersed into a cooling fluid flow. We illustrate the method on some test cases involving three-dimensional time-dependent flow and transport.

Place, publisher, year, edition, pages
Wiley , 2008. Vol. 57, no 9, 1397-1420 p.
Keyword [en]
finite element methods, Navier–Stokes, adaptivity, error estimation, mesh adaptation, advection–diffusion equation
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-19830DOI: 10.1002/fld.1818OAI: oai:DiVA.org:umu-19830DiVA: diva2:207459
Available from: 2009-03-11 Created: 2009-03-11 Last updated: 2011-04-14Bibliographically approved
In thesis
1. Adaptive finite element methods for multiphysics problems
Open this publication in new window or tab >>Adaptive finite element methods for multiphysics problems
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we develop and analyze the performance ofadaptive finite element methods for multiphysics problems. Inparticular, we propose a methodology for deriving computable errorestimates when solving unidirectionally coupled multiphysics problemsusing segregated finite element solvers.  The error estimates are of a posteriori type and are derived using the standard frameworkof dual weighted residual estimates.  A main feature of themethodology is its capability of automatically estimating thepropagation of error between the involved solvers with respect to anoverall computational goal. The a posteriori estimates are used todrive local mesh refinement, which concentrates the computationalpower to where it is most needed.  We have applied and numericallystudied the methodology to several common multiphysics problems usingvarious types of finite elements in both two and three spatialdimensions.

Multiphysics problems often involve convection-diffusion equations for whichstandard finite elements can be unstable. For such equations we formulatea robust discontinuous Galerkin method of optimal order with piecewiseconstant approximation. Sharp a priori and a posteriori error estimatesare proved and verified numerically.

Fractional step methods are popular for simulating incompressiblefluid flow. However, since they are not genuine Galerkin methods, butrather based on operator splitting, they do not fit into the standardframework for a posteriori error analysis. We formally derive an aposteriori error estimate for a prototype fractional step method byseparating the error in a functional describing the computational goalinto a finite element discretization residual, a time steppingresidual, and an algebraic residual.

Place, publisher, year, edition, pages
Umeå: Institutionen för Matematik och matematisk statistik, Umeå universitet, 2009. 171 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 44
Keyword
finite element methods, multiphysics, a posteriori error estimation, duality, adaptivity, discontinuous Galerkin, fractional step methods
Identifiers
urn:nbn:se:umu:diva-30120 (URN)978-91-7264-899-9 (ISBN)
Public defence
2010-01-20, MIT-huset MA 121, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2009-12-18 Created: 2009-12-07 Last updated: 2009-12-18Bibliographically approved
2. Finite element methods for multiscale/multiphysics problems
Open this publication in new window or tab >>Finite element methods for multiscale/multiphysics problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero.

We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

Place, publisher, year, edition, pages
Umeå: Department of Mathematics and Mathematical Statistics, Umeå University, 2011. 26 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 47
Keyword
finite element methods, variational multiscale methods, Galerkin, convergence analysis, multiphysics, a posteriori error estimation, duality, adaptivity
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-42713 (URN)978-91-7459-193-4 (ISBN)
Public defence
2011-05-05, MIT-huset, MA121, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2011-04-14 Created: 2011-04-12 Last updated: 2011-04-14Bibliographically approved

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Larson, Mats GSöderlund, RobertBengzon, Fredrik

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