Finite element methods for parameter dependent problems

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Doctoral thesis (article-based)
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Date
2009
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Language
en
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Verkkokirja (155 KB, 14 s.)
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Research reports / Helsinki University of Technology, Institute of Mathematics. A, 573
Abstract
This thesis develops finite element methods for parameter dependent equations. The interest lies in cases where the nature of the problem and the numerical methods used change with the parameters. The studied examples are the reaction-diffusion problem, the Robin boundary condition, which generalizes the Dirichlet and Neumann conditions, and the Brinkman equation, which generalizes the Stokes and the Darcy equations. The developed methods depend continuously on the problem parameters and work even for the limiting values. A posteriori estimates are derived for all the proposed methods, taking into account the parameters. Significant parts of this work are the implementation of the proposed methods and the numerical verfication of the developed theory.
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Keywords
finite element methods, boundary conditions, Nitsche's method, a posteriori estimation, Stokes equation, Darcy equation, Brinkman equation
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Parts
  • [Publication 1]: Mika Juntunen and Rolf Stenberg. 2008. On a mixed discontinuous Galerkin method. Electronic Transactions on Numerical Analysis, volume 32, pages 17-32.
  • [Publication 2]: Mika Juntunen and Rolf Stenberg. 2009. Nitsche's method for general boundary conditions. Mathematics of Computation, volume 78, number 267, pages 1353-1374. doi:10.1090/S0025-5718-08-02183-2. © 2009 American Mathematical Society (AMS). By permission.
  • [Publication 3]: P. Hansbo and M. Juntunen. 2009. Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. Applied Numerical Mathematics, volume 59, number 6, pages 1274-1289. doi:10.1016/j.apnum.2008.07.003. © 2009 Elsevier Science. By permission.
  • [Publication 4]: Mika Juntunen and Rolf Stenberg. 2009. A residual based a posteriori estimator for the reaction-diffusion problem. Comptes Rendus Mathematique, volume 347, numbers 9-10, pages 555-558. doi:10.1016/j.crma.2009.03.010.
  • [Publication 5]: Mika Juntunen and Rolf Stenberg. 2009. Analysis of finite element methods for the Brinkman problem. Espoo, Finland. 19 pages. Helsinki University of Technology Institute of Mathematics Research Reports, A557. © 2009 by authors.
  • [Publication 6]: Antti Hannukainen, Mika Juntunen, and Rolf Stenberg. 2009. Computations with finite element methods for the Brinkman problem. Espoo, Finland. 19 pages. Helsinki University of Technology Institute of Mathematics Research Reports, A569. © 2009 by authors.
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