BAIL 2006 Book of Abstracts - Institut für Numerische und ...

L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for a
Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of
the Third-Kind Boundary Condition
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A Difference Scheme of Improved Accuracy for a Quasilinear
Singularly Perturbed Elliptic Convection-Diffusion Equation
in the Case of the Third Kind Boundary Condition ∗
Lidia P. Shishkina and Grigory I. Shishkin
Institute of Mathematics and Mechanics,
Ural Branch of Russian Academy of Sciences,
Yekaterinburg 620219, Russia
Lida@convex.ru and shishkin@imm.uran.ru
A boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion
equation on a strip is considered. The third kind boundary condition admitting both Dirichlet
and Neumann conditions is given on the domain boundary. For small values of the perturbation
parameter ε, a boundary layer arises in a neighbourhood of the outflow part of the boundary. For
such a problem, the base (nonlinear) difference scheme constructed by classical approximations
of the problem on piecewise uniform meshes condensing in the layer converges ε-uniformly with
an order of accuracy not higher than 1.
Our aim is for this boundary value problem to construct grid approximations that converge
ε-uniformly with an order of convergence close to two.
Using the�Richardson technique, we construct a (nonlinear) scheme that converges ε-uniformly
at the rate O N −2
1 ln 2 N1+N −2
�
2 , where N1+1 is the number of nodes in the mesh with respect
to x1 and N2+1 is the number of mesh points on a unit interval along the x2-axis. Based on the
nonlinear Richardson scheme, a linearized iterative scheme is constructed where the nonlinear
term is computed using the unknown function taken at the previous iteration. The solution of
this iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of
a geometry progression ε-uniformly with respect to the number of iterations. Thus, the number
of iterations required for solving the problem (as well as the accuracy of the resulting solution)
is independent of the parameter ε.
The use of lower and upper solutions of the linearized iterative Richardson scheme as a
stopping criterion allows us during the computational process to define a current iteration under
which the same ε-uniform accuracy of the solution is achieved as for the nonlinear Richardson
scheme. To construct the improved scheme, the technique developed in [1]–[4] for a Dirichlet
problem is applied.
References
[1] G.I. Shishkin, “The method of increasing the accuracy of solutions of difference schemes
for parabolic equations with a small parameter at the highest derivative”, USSR Comput.
Maths. Math. Phys., 24 (6), 150–157 (1984).
[2] G.I. Shishkin, ”Finite-difference approximations of singularly perturbed elliptic equations”,
Comp. Math. Math. Phys., 38 (12), 1909–1921 (1998).
[3] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singularly
perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030–
1039 (2005).
∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578,
04–01–89007–NWO a) and by the Dutch Research Organisation NWO under grant No. 047.016.008.
Speaker: SHISHKINA, L.P. 49 BAIL 2006
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