### 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; L^{ρ}(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 L^{2}(0, T; L^{ρ}(Ω)) norm is shown for the range 2N/N+1 < ρ < 2. For 2 ≤ ρ < ∞, a rate of O(At1/2ρ) is shown in L^{ρ}(0, T; L^{ρ}(Ω)) norm.

Original language | English |
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Pages (from-to) | 971-989 |

Number of pages | 19 |

Journal | Mathematics of Computation |

Volume | 68 |

Issue number | 227 |

Publication status | Published - 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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## Cite this

^{ρ}error estimates for Galerkin approximations to porous medium and fast diffusion equations.

*Mathematics of Computation*,

*68*(227), 971-989.