Premium
A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations
Author(s) -
Ren Yongqiang,
Cheng Aijie,
Wang Hong
Publication year - 2014
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21790
Subject(s) - mathematics , sobolev space , scaling , mathematical analysis , partial differential equation , norm (philosophy) , finite volume method , a priori and a posteriori , advection , stability (learning theory) , interpolation (computer graphics) , rate of convergence , geometry , animation , philosophy , channel (broadcasting) , physics , computer graphics (images) , electrical engineering , epistemology , machine learning , political science , mechanics , computer science , law , thermodynamics , engineering
We prove an optimal‐order error estimate in a weighted energy norm for finite volume method for two‐dimensional time‐dependent advection–diffusion equations on a uniform space‐time partition of the domain. The generic constants in the estimates depend only on certain norms of the true solution but not on the scaling parameter. These estimates, combined with a priori stability estimates of the governing partial differential equations with full regularity, yield a uniform estimate of the finite volume method, in which the generic constants depend only on the Sobolev norms of the initial and right side data but not on the scaling parameter. We use the interpolation of spaces and stability estimates to derive a uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right‐hand side data. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 17‐43, 2014