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Error Analysis of Extended Discontinuous Galerkin (XdG) Method

Toprakseven, Suayip
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307

Abstract Details

2014, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
The development of efficient numerical methods to approximate solutions of partial differential equation problems that exhibit high frequency oscillations or boundary layers is a challenging task. One promising approach that has gained considerable popularity in the last decade enriches finite element approximations spaces with special functions that capture the difficult solution behavior. This extended finite element method or XFEM is usually coupled with continuous finite elements but several recent papers have enriched the spaces in discontinuous Galerkin framework. Computational results with this extended discontinuous Galerkin method or XdG have been successful and been applied to a wide range of application problems. However, very few theoretical error analyses have been done on either XFEM or, in particular, XdG. Such analyses are provided in this thesis for the XdG method and applied to problems with high frequency solutions and others with boundary layers. Proofs are given showing the XdG approximations are more accurate than those from more standard finite element schemes. These results are provided for elliptic and parabolic problems with solutions that exhibit high frequency oscillations and elliptic problems where boundary layers are present in the solutions. These error estimates are provided in terms of the degree of the polynomials used in the approximation and the largest high frequency or severity of the boundary layer. Computational results for this new method are presented and confirm the theoretical findings.
Donald French, Ph.D. (Committee Chair)
Sookkyung Lim, Ph.D. (Committee Member)
Benjamin Vaughan, Ph.D. (Committee Member)
66 p.
Mathematics
dG; XFEM; XdG

Recommended Citations

Citations

  • Toprakseven, S. (2014). Error Analysis of Extended Discontinuous Galerkin (XdG) Method [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307

    APA Style (7th edition)

  • Toprakseven, Suayip. Error Analysis of Extended Discontinuous Galerkin (XdG) Method. 2014. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307.

    MLA Style (8th edition)

  • Toprakseven, Suayip. "Error Analysis of Extended Discontinuous Galerkin (XdG) Method." Doctoral dissertation, University of Cincinnati, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307

    Chicago Manual of Style (17th edition)

ucin1418733307
584
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This open access ETD is published by University of Cincinnati and OhioLINK.