Projection difference scheme for a parabolic functional differential equation with two-dimensional transformation of arguments

A. V. Razgulin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 10, pp. 1848-1859

Résumé

A nonlinear parabolic functional differential equation with the functional part containing a generalized superposition of the unknown solution and a transformation of the two-dimensional spatial argument is considered. A projection difference scheme for the approximation of the initial Dirichlet boundary value problem in a rectangle is proposed for a wide class of measurable, including noninvertible, transformations. An estimate of the rate of convergence to the generalized solutions of the initial problem of order $O(\tau^{1/4-\gamma}+h^{1/2-2\gamma})$ in the norm $L_2(Q)$ without a priori assumptions on the invertibility of the transformation and without any mesh size matching is obtained.

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