### 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 |
---|---|

Pages (from-to) | 971-989 |

Number of pages | 19 |

Journal | Mathematics of Computation |

Volume | 68 |

Issue number | 227 |

Publication status | Published - Jul 1999 |

Externally published | Yes |

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### 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
- Applied Mathematics
- Computational Mathematics

### Cite this

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

*Mathematics of Computation*,

*68*(227), 971-989.

**A priori L ^{ρ} error estimates for Galerkin approximations to porous medium and fast diffusion equations.** / Wei, Dongming; Lefton, Lew.

Research output: Contribution to journal › Article

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

*Mathematics of Computation*, vol. 68, no. 227, pp. 971-989.

^{ρ}error estimates for Galerkin approximations to porous medium and fast diffusion equations. Mathematics of Computation. 1999 Jul;68(227):971-989.

}

TY - JOUR

T1 - A priori Lρ error estimates for Galerkin approximations to porous medium and fast diffusion equations

AU - Wei, Dongming

AU - Lefton, Lew

PY - 1999/7

Y1 - 1999/7

N2 - 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 2(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.

AB - 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 2(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.

KW - Cauchy-Dirichlet problem

KW - Fast diffusion equation

KW - Finite elements

KW - Galerkin approximations

KW - L error estimates

KW - Porous medium equation

UR - http://www.scopus.com/inward/record.url?scp=0039390040&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039390040&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0039390040

VL - 68

SP - 971

EP - 989

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 227

ER -