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

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

num.math.uni.goettingen.de
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G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularly perturbed quasilinear parabolic convection-diffusion equations ✬ ✫ A posteriori adapted meshes in the approximation of singularly perturbed quasilinear parabolic convection-diffusion equations ∗ Grigory I. Shishkin A Dirichlet problem on a segment for a quasilinear parabolic convection-diffusion equation with a small (perturbation) parameter ε multiplying the highest derivative is considered. For this problem, a solution of the classical finite difference scheme on a uniform mesh converges only under the condition h ≪ ε, where h is the step-size of the space mesh; moreover, the order of convergence in x is O � ε N −1� , where N + 1 is the number of nodes in the uniform mesh with respect to x. To improve the accuracy of the approximate solution, we apply a posteriori sequential procedure of grid refinement in the subdomains that are defined by the gradient of solutions of intermediate discrete problems. The correction of the grid solutions is performed only on these subdomains, where uniform meshes are used. We construct a difference scheme that converges ”almost ε-uniformly”, i.e., with an error weakly depending on the parameter ε. � The convergence rate of the constructed scheme is O where N1 + 1 and N0 + 1 are the numbers of mesh points with respect to x and t, respectively, ν is an arbitrary number from (0, 1]. Thus, the scheme on a posteriori adapted meshes converges under the condition N −1 ≪ εν , which is essentially weaker in comparison with the scheme on uniform meshes. ε −ν N −1 1 + N −1/2 1 ∗ This research was supported in part by the Russian Foundation for Basic Research (grant No 04-01-00578, 04–01–89007–NWO a), by the Dutch Research Organisation NWO under grant No 047.016.008 and by the Boole Centre for Research in Informatics, National University of Ireland, Cork. 2 + N −1 � 0 , Speaker: SHISHKIN, G.I. 16 BAIL 2006 ✩ ✪

G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularly perturbed quasilinear parabolic convection-diffusion equations ✬ ✫ A scheme on a posteriori adapted meshes in the case of a linear problem is considered in [1]. References [1] G.I. Shishkin, A posteriori adapted (to the solution gradient) grids in the approximation of singularly perturbed convection-diffusion equations, Vychisl. Tekhnol. (Computational Technologies), 6 (1), 72-87 (2001) (in Russian). Speaker: SHISHKIN, G.I. 17 BAIL 2006 3 ✩ ✪

G.I. SHISHKIN: A posteriori adapted meshes in the approximation <strong>of</strong> singularly<br />

perturbed quasilinear parabolic convection-diffusion equations<br />

✬<br />

✫<br />

A scheme on a posteriori adapted meshes in the case <strong>of</strong> a<br />

linear problem is considered in [1].<br />

References<br />

[1] G.I. Shishkin, A posteriori adapted (to the solution gradient)<br />

grids in the approximation <strong>of</strong> singularly perturbed<br />

convection-diffusion equations, Vychisl. Tekhnol. (Computational<br />

Technologies), 6 (1), 72-87 (2001) (in Russian).<br />

Speaker: SHISHKIN, G.I. 17 <strong>BAIL</strong> <strong>2006</strong><br />

3<br />

✩<br />

✪

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