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Adaptive piecewise constant discontinuous Galerkin methods for convection-diffusion 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. (UMIT)
2009 (English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we present a discontinuous Galerkin method with  piecewise constant approximation for convection-diffusion type  equations. We show that if the discretization is carefully chosen, then the method is optimal in the L2 norm as well as in a  discrete energy norm measuring the normal flux across element  boundaries. We also derive a posteriori error estimates and  illustrate their effectiveness in combination with adaptive mesh  refinement on a few benchmark problems.

Place, publisher, year, edition, pages
National Category
URN: urn:nbn:se:umu:diva-30254OAI: diva2:281193
Available from: 2009-12-15 Created: 2009-12-15 Last updated: 2012-01-16Bibliographically 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.
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 44
finite element methods, multiphysics, a posteriori error estimation, duality, adaptivity, discontinuous Galerkin, fractional step methods
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)
Available from: 2009-12-18 Created: 2009-12-07 Last updated: 2009-12-18Bibliographically approved

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Bengzon, FredrikLarson, Mats G
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