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

Dongming Wei; Lew Lefton

A priori L^ρ error estimates for Galerkin approximations to porous medium and fast diffusion equations

Research output: Contribution to journal › Article

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 L²(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

Fingerprint

Fast Diffusion Equation

Galerkin Approximation

Porous Media

Porous materials

Error Estimates

Norm

Fast Diffusion

Porous Medium Equation

Finite Difference Approximation

Convex Domain

Generalized Equation

Global Convergence

Range of data

Dirichlet Problem

Convergence Rate

Euler

Bounded Domain

Exact Solution

Converge

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

Wei, D., & Lefton, L. (1999). A priori L^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.

In: Mathematics of Computation, Vol. 68, No. 227, 07.1999, p. 971-989.

Research output: Contribution to journal › Article

Wei, D & Lefton, L 1999, 'A priori L^ρ error estimates for Galerkin approximations to porous medium and fast diffusion equations', Mathematics of Computation, vol. 68, no. 227, pp. 971-989.

Wei D, Lefton L. A priori L^ρ error estimates for Galerkin approximations to porous medium and fast diffusion equations. Mathematics of Computation. 1999 Jul;68(227):971-989.

Wei, Dongming ; Lefton, Lew. / A priori L^ρ error estimates for Galerkin approximations to porous medium and fast diffusion equations. In: Mathematics of Computation. 1999 ; Vol. 68, No. 227. pp. 971-989.

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