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Adaptive finite element methods for multiphysics problems
Umeå University, Faculty of Science and Technology, Mathematics and Mathematical Statistics. (computational mathematics)
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 [en]
finite element methods, multiphysics, a posteriori error estimation, duality, adaptivity, discontinuous Galerkin, fractional step methods
Identifiers
URN: urn:nbn:se:umu:diva-30120ISBN: 978-91-7264-899-9 (print)OAI: oai:DiVA.org:umu-30120DiVA: diva2:281257
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
List of papers
1. Adaptive finite element approximation of multiphysics problems
Open this publication in new window or tab >>Adaptive finite element approximation of multiphysics problems
2007 (English)In: Communications in Numerical Methods in Engineering, ISSN 1069-8299, E-ISSN 1099-0887, Vol. 24, no 6, 505-521 p.Article in journal (Refereed) Published
Abstract [en]

Simulation of multiphysics problems is a common task in applied research and industry. Often a multiphysics solver is built by connecting several single-physics solvers into a network. In this paper, we develop a basic adaptive methodology for such multiphysics solvers. The adaptive methodology is based on a posteriori error estimates that capture the influence of the discretization errors in the different solvers on a given functional output. These estimates are derived using duality-based techniques.

Place, publisher, year, edition, pages
Wiley InterScience, 2007
Keyword
multiphysics problems, error estimation, MEMS
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:umu:diva-8091 (URN)10.1002/cnm.1087 (DOI)000257339200008 ()
Available from: 2008-01-14 Created: 2008-01-14 Last updated: 2017-12-14Bibliographically approved
2. Simulation of multiphysics problems using adaptive finite elements
Open this publication in new window or tab >>Simulation of multiphysics problems using adaptive finite elements
2006 (English)In: Applied parallel computing state of the art in scientific computing: 8th International Workshop, PARA 2006, Umeå, Sweden, umeå: department of Mathematics, Umeå University , 2006, 1-14 p.Conference paper, Published paper (Refereed)
Abstract [en]

Real world applications often involve several types of physics. In practice, one often solves such multiphysics problems by using already existing single physics solvers. To satisfy an overall accuracy, it is critical to understand how accurate the individual single physics solution must be. In this paper we present a framework for a posteriori error estimation of multiphysics problems and derive an algorithm for estimating the total error. We illustrate the technique by solving a coupled flow and transport problem with application in porous media flow.

Place, publisher, year, edition, pages
umeå: department of Mathematics, Umeå University, 2006
Identifiers
urn:nbn:se:umu:diva-8112 (URN)
Conference
8th International Workshop, PARA 2006, Umeå, Sweden
Available from: 2008-01-15 Created: 2008-01-15 Last updated: 2010-05-11Bibliographically approved
3. 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: 2017-12-13Bibliographically approved
4. Adaptive finite element approximation of multiphysics problems: a fluid structure interaction model problem
Open this publication in new window or tab >>Adaptive finite element approximation of multiphysics problems: a fluid structure interaction model problem
2010 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 84, no 12, 1451-1465 p.Article in journal (Refereed) Published
Abstract [en]

We consider computation of the displacement of an elastic object immersed into a viscous incompressible flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. We derive an a posteriori error estimate for this one way coupled problem using duality techniques. Based on these estimates we develop an adaptive algorithm that automatically constructs a suitable adapted mesh for the fluid and solid domains given goal quantities specified on the solid problem.

Keyword
fluid-structure interaction, finite element methods, adaptivity, error estimation
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:umu:diva-30228 (URN)10.1002/nme.2945 (DOI)000285261100002 ()
Available from: 2009-12-14 Created: 2009-12-14 Last updated: 2017-12-12Bibliographically approved
5. Adaptive piecewise constant discontinuous Galerkin methods for convection-diffusion problems
Open this publication in new window or tab >>Adaptive piecewise constant discontinuous Galerkin methods for convection-diffusion problems
2009 (English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we present a discontinuous Galerkin method with  piecewise constant approximation for convection-diffusion type  equations. We show that if the discretization is carefully chosen, then the method is optimal in the L2 norm as well as in a  discrete energy norm measuring the normal flux across element  boundaries. We also derive a posteriori error estimates and  illustrate their effectiveness in combination with adaptive mesh  refinement on a few benchmark problems.

National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-30254 (URN)
Available from: 2009-12-15 Created: 2009-12-15 Last updated: 2012-01-16Bibliographically approved
6. A posteriori error estimates for fractional step methods in fluid mechanics
Open this publication in new window or tab >>A posteriori error estimates for fractional step methods in fluid mechanics
2009 (English)In: Computational Methods in Marine Engineering / [ed] P. Bergan, J. Garcia, E. Onate and T. Kvamsdal, 2009Conference paper, Published paper (Refereed)
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-30226 (URN)
Conference
CIMNE, Barcelona, Spain
Available from: 2009-12-14 Created: 2009-12-14 Last updated: 2013-08-12Bibliographically approved

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