A priori Lρ error estimates for Galerkin approximations to porous medium and fast diffusion equations

Dongming Wei, Lew Lefton

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Galerkin approximations to solutions of a Caucliy-Dirichlet problem governed by the generalized porous medium equation (formula presented) on bounded convex domains are considered. The range of the parameter ρ includes the fast diffusion case 1 < ρ < 2. Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in L(0, T; Lρ(Ω)) norm with an error controlled by O(Δt1/1) for 1 < p < 2 and O(Δt1/2ρ) for 2 ≤ p < ∞. For the fully discrete problem, a global convergence rate of O(Δt1/4) in L2(0, T; Lρ(Ω)) norm is shown for the range 2N/N+1 < ρ < 2. For 2 ≤ ρ < ∞, a rate of O(At1/2ρ) is shown in Lρ(0, T; Lρ(Ω)) norm.

Original languageEnglish
Pages (from-to)971-989
Number of pages19
JournalMathematics of Computation
Volume68
Issue number227
Publication statusPublished - Jul 1 1999

Keywords

  • Cauchy-Dirichlet problem
  • Fast diffusion equation
  • Finite elements
  • Galerkin approximations
  • L error estimates
  • Porous medium equation

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

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