A hybridizable discontinuous Galerkin method for the p-Laplacian

Bernardo Cockburn, Jiguang Shen

Research output: Contribution to journalArticlepeer-review

29 Scopus citations

Abstract

We propose the first hybridizable discontinuous Galerkin method for the p-Laplacian equation. When using polynomials of degree k ≥ 0 for the approximation spaces of u, ∇u, and |∇u|p-2∇u, the method exhibits optimal k + 1 order of convergence for all variables in L1- and Lp-norms in our numerical experiments. For k ≥ 1, an elementwise computation allows us to obtain a new approximation uh∗ that converges to u with order k + 2. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of u on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of using frozen-coefficient preconditioners, and indicating that the second method is superior to the first one even though it has more unknowns.

Original languageEnglish (US)
Pages (from-to)A545-A566
JournalSIAM Journal on Scientific Computing
Volume38
Issue number1
DOIs
StatePublished - 2016

Keywords

  • Degenerate elliptic equations
  • Discontinuous Galerkin methods
  • Hybridization
  • Nonlinear minimization algorithms
  • Postprocessing
  • Preconditioning
  • p-Laplacian operator

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