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Simulation of multiphysics problems using adaptive finite elements
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.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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. 1-14 p.
Identifiers
URN: urn:nbn:se:umu:diva-8112OAI: oai:DiVA.org:umu-8112DiVA: diva2:147783
Conference
8th International Workshop, PARA 2006, Umeå, Sweden
Available from: 2008-01-15 Created: 2008-01-15 Last updated: 2010-05-11Bibliographically 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. Duality-based adaptive finite element methods with application to time-dependent problems
Open this publication in new window or tab >>Duality-based adaptive finite element methods with application to time-dependent problems
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

To simulate real world problems modeled by differential equations, it is often not sufficient to  consider and tackle a single equation. Rather, complex phenomena are modeled by several partial dierential equations that are coupled to each other. For example, a heart beat involve electric activity, mechanics of the movement of the walls and valves, as well as blood fow - a true multiphysics problem. There may also be ordinary differential equations modeling the reactions on a cellular level, and these may act on a much finer scale in both space and time. Determining efficient and accurate simulation tools for such multiscalar multiphysics problems is a challenge.

The five scientific papers constituting this thesis investigate and present solutions to issues regarding accurate and efficient simulation using adaptive finite element methods. These include handling local accuracy through submodeling, analyzing error propagation in time-dependent  multiphysics problems, developing efficient algorithms for adaptivity in time and space, and deriving error analysis for coupled PDE-ODE systems. In all these examples, the error is analyzed and controlled using the framework of dual-weighted residuals, and the spatial meshes are handled using octree based data structures. However, few realistic geometries fit such grid and to address this issue a discontinuous Galerkin Nitsche method is presented and analyzed.

Place, publisher, year, edition, pages
Umeå: Institutionen för matematik och matematisk statistik, Umeå universitet, 2010. 37 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 45
Keyword
finite element methods, dual-weighted residual method, multiphysics, a posteriori error estimation, adaptive algorithms, discontinuous Galerkin
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-33872 (URN)978-91-7459-023-4 (ISBN)
Public defence
2010-06-10, MIT-huset, MA121, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2010-05-11 Created: 2010-05-07 Last updated: 2010-05-24Bibliographically approved

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Bengzon, FredrikJohansson, AugustLarson, Mats GSöderlund, Robert

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