On Convergence of Difference Schemes for Dirichlet IBVP for Two-Dimensional Quasilinear Parabolic Equations with Mixed Derivatives and Generalized Solutions

Abstract

Abstract For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations with mixed derivatives monotone linearized difference scheme is constructed. The ellipticity conditions c 1 ⁢ ∑ α = 1 2 ξ α 2 ≤ ∑ α , β = 1 2 k α ⁢ β ⁢ ( u ) ⁢ ξ α ⁢ ξ β ≤ c 2 ⁢ ∑ α = 1 2 ξ α 2 c_{1}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2}\leq\sum_{\alpha,\beta=1}^{2}k_{\alpha% \beta}(u)\xi_{\alpha}\xi_{\beta}\leq c_{2}\sum_{\alpha=1}^{2}\xi_{\alpha}^{2} are assumed to be fulfilled for the sign alternating solution u ⁢ ( 𝐱 , t ) ∈ D ¯ ⁢ ( u ) {u(\mathbf{x},t)\in\bar{D}(u)} only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle, not only two-sided estimates for the approximate solution y but also its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivatives ∂ ⁡ u ∂ ⁡ x i {\frac{\partial u}{\partial x_{i}}} have discontinuities of the first kind in the neighborhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. The convergence of an approximate solution to a generalized solution of a differential problem in the grid norm L 2 {L_{2}} is proved.