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Finite element methods for multiscale/multiphysics problems
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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 [en]
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: urn:nbn:se:umu:diva-42713ISBN: 978-91-7459-193-4OAI: oai:DiVA.org:umu-42713DiVA: diva2:410033
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
List of papers
1. Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer
Open this publication in new window or tab >>Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer
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
Keyword
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:nbn:se:umu:diva-19830 (URN)10.1002/fld.1818 (DOI)
Available from: 2009-03-11 Created: 2009-03-11 Last updated: 2011-04-14Bibliographically approved
2. Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions
Open this publication in new window or tab >>Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions
2010 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 4, 781-795 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.

Place, publisher, year, edition, pages
Springer, 2010
Keyword
Finite element methods, Joule heating problem, Convergence analysis
Identifiers
urn:nbn:se:umu:diva-42700 (URN)10.1007/s10543-010-0287-z (DOI)000284594700006 ()
Available from: 2011-04-12 Created: 2011-04-12 Last updated: 2012-01-14Bibliographically approved
3.
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4. A variational multiscale method for poissons equation on mixed form
Open this publication in new window or tab >>A variational multiscale method for poissons equation on mixed form
(English)Manuscript (preprint) (Other academic)
Identifiers
urn:nbn:se:umu:diva-42709 (URN)
Available from: 2011-04-12 Created: 2011-04-12 Last updated: 2011-04-14Bibliographically approved
5. A discontinuous galerkin multiscale method for first order hyperbolic equations
Open this publication in new window or tab >>A discontinuous galerkin multiscale method for first order hyperbolic equations
(English)Manuscript (preprint) (Other academic)
Identifiers
urn:nbn:se:umu:diva-42711 (URN)
Available from: 2011-04-12 Created: 2011-04-12 Last updated: 2011-04-14Bibliographically approved

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Söderlund, Robert
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