Difference Schemes For Convection-diffusion Equations With Generalized Solutions

High-accuracy difference schemes are proposed for a multidimensional convection-diffusion equation in classes of generalized solutions. The piecewise polynomial finite element method is used to approximate spatial variables and the resulting Cauchy problem for a system of ordinary differential equations. As a result, third-order accuracy schemes with respect to spatial variables and fourth-order accuracy schemes with respect to time are obtained. Stability conditions and a priori estimates for the solution of the scheme in classes of nonsmooth solutions are obtained. Based on a special technique, convergence theorems were established in a weaker functional norm. Convergence and accuracy theorems are proven.

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