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Adaptive finite element approximation of multiphysics problems
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.
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. Vol. 24, no 6, 505-521 p.
Keyword [en]
multiphysics problems, error estimation, MEMS
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-8091DOI: 10.1002/cnm.1087ISI: 000257339200008OAI: oai:DiVA.org:umu-8091DiVA: diva2:147762
Available from: 2008-01-14 Created: 2008-01-14 Last updated: 2017-12-14Bibliographically 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

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