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Adaptive finite element solution of coupled PDE-ODE systems
Department of Mathematics, Colorado State University, Fort collins, CO 80523.
Department of Mathematics and Department of Statistics, Colorado State University, Fort collins, CO 80523.
Department of Mathematics, University of Wyoming, laramie, WY 82071.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider an implicit / explicit method for solving a semilinear parabolic partial differential equation (PDE) coupled to a set of nonlinear ordinary differential equations (ODEs). More specifically the PDE of interest is the heat equation where the right hand side couple with the ODEs. For this system, a posteriori error estimates are derived using the method of dual-weighted residuals giving indicators useful for constructing adaptive algorithms.

We distinguish the errors in time and space for the PDE and the ODEs separately and include errors due to transferring the solutions between the equations. In addition, since the ODEs in many applications are defined on a much smaller spatial scale than what can be resolved by the finite element discretization for the PDE, the error terms include possible projection errors arising when transferring the global PDE solution onto the local ODEs. Recovery errors due to passing the local ODE solutions to the PDE are also included in this analysis.

The method is illustrated on a realistic problem consisting of a semilinear PDE and a set of nonlinear ODEs modeling the electrical activity in the heart. The method is computationally expensive, why an adaptive algorithm using blocks is used.

National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-33866OAI: oai:DiVA.org:umu-33866DiVA: diva2:318483
Available from: 2010-05-07 Created: 2010-05-07 Last updated: 2010-05-24Bibliographically approved
In thesis
1. 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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Johansson, AugustLarson, Mats G.

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CiteExportLink to record
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Citation style
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