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A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0001-7838-1307
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2017 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 308, 2613-2649 p.Article in journal (Refereed) Published
Abstract [en]

We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R-3. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in L-2 norm. This can be seen as an extension of the formalism and method originally used by Dziuk ( 1988) for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation inverted right perpendicular(h) of an implicitly defined surface inverted right perpendicular we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on inverted right perpendicular. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.

Place, publisher, year, edition, pages
2017. Vol. 86, no 308, 2613-2649 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-79207DOI: 10.1090/mcom/3179ISI: 000404567600003OAI: oai:DiVA.org:umu-79207DiVA: diva2:640280
Note

Originally published in manuscript form with title [A continuous/discontinuous Galerkin method for the biharmonic problem on surfaces]

Available from: 2013-08-13 Created: 2013-08-13 Last updated: 2017-10-16Bibliographically approved
In thesis
1. Finite Element Methods for Thin Structures with Applications in Solid Mechanics
Open this publication in new window or tab >>Finite Element Methods for Thin Structures with Applications in Solid Mechanics
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Thin and slender structures are widely occurring both in nature and in human creations. Clever geometries of thin structures can produce strong constructions while requiring a minimal amount of material. Computer modeling and analysis of thin and slender structures have their own set of problems, stemming from assumptions made when deriving the governing equations. This thesis deals with the derivation of numerical methods suitable for approximating solutions to problems on thin geometries. It consists of an introduction and four papers.

In the first paper we introduce a thread model for use in interactive simulation. Based on a three-dimensional beam model, a corotational approach is used for interactive simulation speeds in combination with adaptive mesh resolution to maintain accuracy.

In the second paper we present a family of continuous piecewise linear finite elements for thin plate problems. Patchwise reconstruction of a discontinuous piecewise quadratic deflection field allows us touse a discontinuous Galerkin method for the plate problem. Assuming a criterion on the reconstructions is fulfilled we prove a priori error estimates in energy norm and L2-norm and provide numerical results to support our findings.

The third paper deals with the biharmonic equation on a surface embedded in R3. We extend theory and formalism, developed for the approximation of solutions to the Laplace-Beltrami problem on an implicitly defined surface, to also cover the biharmonic problem. A priori error estimates for a continuous/discontinuous Galerkin method is proven in energy norm and L2-norm, and we support the theoretical results by numerical convergence studies for problems on a sphere and on a torus.

In the fourth paper we consider finite element modeling of curved beams in R3. We let the geometry of the beam be implicitly defined by a vector distance function. Starting from the three-dimensional equations of linear elasticity, we derive a weak formulation for a linear curved beam expressed in global coordinates. Numerical results from a finite element implementation based on these equations are compared with classical results.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2013. vi, 18 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 54
Keyword
a priori error estimation, finite element method, discontinuous Galerkin, corotation, Kirchhoff-Love plate, curved beam, biharmonic equation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-79297 (URN)978-91-7459-653-3 (ISBN)978-91-7459-654-0 (ISBN)
Public defence
2013-09-06, S205h, Samhällsvetarhuset, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2013-08-16 Created: 2013-08-13 Last updated: 2013-08-16Bibliographically approved

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Publisher's full textarXiv:1305.2740

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Larsson, KarlLarson, Mats G

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