Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems. Bi, C. & Ginting, V. Journal of Scientific Computing, 49(3):311-331, 2011.

Paper doi abstract bibtex

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r&ge1 for a class of quasi-linear elliptic problems in . We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ^d, d=2,3 and use it to establish the convergence of the two-grid method for problems in &Omega &subset ℝ³.

@article {MR2853153,
    AUTHOR = {Bi, C. and Ginting, V.},
     TITLE = {Two-{G}rid {D}iscontinuous {G}alerkin {M}ethod for {Q}uasi-{L}inear
              {E}lliptic {P}roblems},
  JOURNAL = {Journal of Scientific Computing},
    VOLUME = {49},
      YEAR = {2011},
    NUMBER = {3},
     PAGES = {311-331},
      ISSN = {0885-7474},
     CODEN = {JSCOEB},
   MRCLASS = {65N30 (65N12)},
  MRNUMBER = {2853153 (2012m:65406)},
MRREVIEWER = {Alexandre L. Madureira},
       DOI = {10.1007/s10915-011-9463-9},
       URL = {http://dx.doi.org/10.1007/s10915-011-9463-9},
       ABSTRACT="In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree <i>r</i>&ge1 for a class of quasi-linear elliptic problems in . We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken <i>H</i><sup>1</sup>-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in \mathbb{R}<sup><i>d</i></sup>, <i>d</i>=2,3 and use it to establish the convergence of the two-grid method for problems in
       &Omega &subset \mathbb{R}<sup>3</sup>."
}

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