A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations. Issue 1 (18th May 2013)

• Record Type: Journal Article
• Authors: Ren, Yongqiang; Cheng, Aijie; Wang, Hong
• Is Part Of: Numerical methods for partial differential equations. Volume 30:Issue 1(2014:Jan.)
• Page Start: 17
• Page End: 43
• Publication Date: 2013-05-18
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21790
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>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 <italic>a priori</italic> 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</p> </abstract>