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 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
A priori Lρ 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
}
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 -