A uniform numerical method for quasilinear singular perturbation problems without turning points

Maximum Norm A Posteriori Error Estimates for a One-Dimensional Convection-Diffusion Problem

“…15e-3 1.32e-1 6.94e-2 9.26e-2 1.11e-1 8.37e-1 9.29e-2 1.11e-1 8.41e-1 8.41e-1 1.04 0.95 0.99 0.88 0.99 0.88 64 4.46e-3 6.81e-2 6.55e-2 4.65e-2 6.02e-2 7.72e-1 4.67e-2 6.02e-2 7.75e-1 7.75e-1 1.02 0.98 1.00 0.94 1.00 0.94 128 2.20e-3 3.46e-2 6.36e-2 2.33e-2 3.14e-2 7.41e-1 2.34e-2 3.14e-2 7.44e-1 7.44e-1 1.01 0.99 1.00 0.97 1.00 0.97 256 1.09e-3 1.74e-2 6.27e-2 1.17e-2 1.61e-2 7.25e-1 1.17e-2 1.61e-2 7.28e-1 7.28e-1 1.00 0.99 1.00 0.98 1.00 0.98 512 5.45e-4 8.75e-3 6.22e-2 5.83e-3 8.12e-3 7.18e-1 5.85e-3 8.11e-3 7.21e-1 7.21e-1 see [17,21,31]. Note that our numerical results for the quasi-linear problem will be compared with the asymptotic solution u A (x), and since u(x) ∈ [−π/2, 0], we can take β = e −π/2 .…”

“…where ε ∈ (0, 1] is a small parameter, and b u (x, u) has a positive lower bound, which implies that the solution u(x) generally has an exponential boundary layer at x = 0 [18,31]. As a result, the accuracy of classical numerical methods depends not only on the number of the meshnodes N but also on the parameter ε (e.g., for (1.1) the error depends on N/ε) [22,26].…”

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

“…Nonlinear problems of more general or different type than the problem (1)-(2) were studied in e.g. [5,12,17,30,31]. A large number of difference schemes belongs to the group of exponentially fitted schemes or their uniformly convergent versions.…”

On numerical solution of a mildly nonlinear turning point problem

“…It is easy to dérive analogous properties of the function X in ƒ". The same function X was used in [18] and a very similar in [17] (see mesh generating fonctions in [15], [16] and [20] as well). Essentially, the part w is a certain modification of the inverse of the interior layer function V z {x) for x 5= 0 (cf.…”

“…We shall allow a mild nonlinearity of the problem and show that properties of the continuous solution may be used in order to obtain numerical methods uniform in E. We shall follow the technique from [18] {cf. [15], [16], [17], [20] and [9], [10] as well) which uses fînite-difference schemes on special non-equidistant meshes. In this paper we shall be interested in the uniformity in e, rather than in high accuracy.…”