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Duality-based adaptive finite element methods with application to time-dependent problems
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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 [en]
finite element methods, dual-weighted residual method, multiphysics, a posteriori error estimation, adaptive algorithms, discontinuous Galerkin
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-33872ISBN: 978-91-7459-023-4 (print)OAI: oai:DiVA.org:umu-33872DiVA: diva2:318503
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
List of papers
1. Adaptive submodeling for linear elasticity problems with multiscale geometric features
Open this publication in new window or tab >>Adaptive submodeling for linear elasticity problems with multiscale geometric features
2005 (English)In: Multiscale Methods in Science and Engineering / [ed] Björn Engquist, Olof Runborg and Per Lötstedt, Berlin Heidelberg: Springer Verlag , 2005, Vol. 44, 169-180 p.Chapter in book (Other academic)
Abstract [en]

Submodeling is a procedure for local enhancement of the resolution of a coarse global finite element solution by solving a local problem on a subdomain containing an area of particular interest. We focus on linear elasticity and computation of local stress levels determined by the local geometry of the domain. We derive a posteriori error estimates for the submodeling procedure using duality techniques. Based on these estimates we propose an adaptive procedure for automatic choice of the resolution and size of the submodel. The procedure is illustrated for problems of industrial interest.

Place, publisher, year, edition, pages
Berlin Heidelberg: Springer Verlag, 2005
Series
Lecture notes in Computational Science and Engineering, ISSN 1439-7358 ; 44
Keyword
adaptive multiscale method, a posteriori error estimate, finite element, meshrefinement
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-33869 (URN)10.1007/b137594 (DOI)978-3-540-25335-8 (Print) 978-3-540-26444-6 (Online) (ISBN)
Available from: 2010-05-07 Created: 2010-05-07 Last updated: 2010-05-24Bibliographically 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. Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions
Open this publication in new window or tab >>Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions
Show others...
2010 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 32, no 4, 2121-2145 p.Article in journal (Refereed) Published
Abstract [en]

We describe and test an adaptive algorithm for evolution problems that employs a sequence of "blocks" consisting of fixed, though non-uniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, re-meshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue whith a blockadaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies to achieve this goal using adjoint-based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2010
Keyword
a posteriori error analysis, adaptive error control, adaptive mesh refinement, adjoint problem, discontinuous Galerkin method, duality, generalized Green's function, goal oriented error estiamtes, residual, variational analysis
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-33871 (URN)10.1137/090753826 (DOI)000280771100020 ()
Available from: 2010-05-07 Created: 2010-05-07 Last updated: 2017-12-12Bibliographically approved
4. Adaptive finite element solution of coupled PDE-ODE systems
Open this publication in new window or tab >>Adaptive finite element solution of coupled PDE-ODE systems
Show others...
(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:nbn:se:umu:diva-33866 (URN)
Available from: 2010-05-07 Created: 2010-05-07 Last updated: 2010-05-24Bibliographically approved
5. A discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary
Open this publication in new window or tab >>A discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present a discountinous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. We also discuss the implementation of the method and present a numerical example in three dimensions.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-33868 (URN)
Available from: 2010-05-07 Created: 2010-05-07 Last updated: 2010-05-24Bibliographically approved

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