TY - GEN

T1 - On a necessary requirement for re-uniform numerical methods to solve boundary layer equations for flow along a flat plate

AU - Shishkin, Grigorii I.

AU - Farrell, Paul A.

AU - Hegarty, Alan F.

AU - Miller, John J.H.

AU - O’Riordan, Eugene

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001.

PY - 2001

Y1 - 2001

N2 - We consider grid approximations of a boundary value problem for the boundary layer equations modeling flow along a flat plate in a region excluding a neighbourhood of the leading edge. The problem is singularly perturbed with the perturbation parameter ε = 1/Re multiplying the highest derivative. Here the parameter ε takes any values from the half-interval (0,1], and Re is the Reynolds number. It would be of interest to construct an Re-uniform numerical method using the simplest grids, i.e., uniform rectangular grids, that could provide effective computational methods. To this end, we are free to use any technique even up to fitted operator methods, however, with fitting factors independent of the problem solution. We show that for the Prandtl problem, even in the case when its solution is self-similar, there does not exist a fitted operator method that converges Re-uniformly. Thus, combining a fitted operator and uniform meshes, we do not succeed in achieving Re-uniform convergence. Therefore, the use of the fitted mesh technique, based on meshes condensing in a parabolic boundary layer, is a necessity in constructing Re-uniform numerical methods for the above class of flow problems.

AB - We consider grid approximations of a boundary value problem for the boundary layer equations modeling flow along a flat plate in a region excluding a neighbourhood of the leading edge. The problem is singularly perturbed with the perturbation parameter ε = 1/Re multiplying the highest derivative. Here the parameter ε takes any values from the half-interval (0,1], and Re is the Reynolds number. It would be of interest to construct an Re-uniform numerical method using the simplest grids, i.e., uniform rectangular grids, that could provide effective computational methods. To this end, we are free to use any technique even up to fitted operator methods, however, with fitting factors independent of the problem solution. We show that for the Prandtl problem, even in the case when its solution is self-similar, there does not exist a fitted operator method that converges Re-uniformly. Thus, combining a fitted operator and uniform meshes, we do not succeed in achieving Re-uniform convergence. Therefore, the use of the fitted mesh technique, based on meshes condensing in a parabolic boundary layer, is a necessity in constructing Re-uniform numerical methods for the above class of flow problems.

UR - http://www.scopus.com/inward/record.url?scp=84944128414&partnerID=8YFLogxK

U2 - 10.1007/3-540-45262-1_85

DO - 10.1007/3-540-45262-1_85

M3 - Conference contribution

AN - SCOPUS:84944128414

SN - 9783540418146

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 723

EP - 731

BT - Numerical Analysis and Its Applications - 2nd International Conference, NAA 2000, Revised Papers

A2 - Vulkov, Lubin

A2 - Yalamov, Plamen

A2 - Waniewski, Jerzy

PB - Springer Verlag

T2 - 2nd International Conference on Numerical Analysis and Its Applications, NAA 2000

Y2 - 11 June 2000 through 15 June 2000

ER -