On an approximate solution method for the problem of surface and groundwater combined movement with exact approximation on the section line

In this paper, the initial-boundary problem for two nonlinear parabolic combined equations has been considered. One of the equations is set on the bounded domain $\Omega\subset R^2,$ another equation is set along the curve lying in $\overline\Omega$. Both of the equations are parabolic equations with double degeneration. The degeneration can be present at the space operator. Furthermore, the nonlinear function which is under the sign of partial derivative with respect to the variable $t$, can be bound to zero. This problem has an applied character: such structure is needed to describe the process of surface and ground water combined movement. In this case, the desired function determines the level of water above the given impenetrable bottom, the section simulates the riverbed. The Bussinesk equation has been used for mathematical description of the groundwater filtration process in the domain $\Omega$; a diffusion analogue of the Saint-Venant's system has been used on the section for description of the process of water level change in the open channel. Earlier, the authors proved the theorems of generalized solution existence and uniqueness for the considered problem from the functions classes which are called strengthened Sobolev spaces in the literature. To obtain these results, we used the technique which was created by the German mathematicians (H. W. Alt, S. Luckhaus, F. Otto) to establish the correctness of the problems with a double degeneration.
In this paper, we have proposed and investigated an approximate solution method for the above-stated problem. This method has been constructed using semidiscretization with respect to the variable $t$ and the finite element method for space variables. Triangulation of the domain has been accomplished by triangles. The mesh has been set on the section line. On each segment of the line section lying between the nearby mesh points, on both side of this segment we have constructed the triangles with a common side which matches with the picked segment of the section line. Triangulation of the rest of domain has been accomplished by triangles as commonly accepted. A list of a priori estimates has been obtained. The convergence of the constructed method has been proved.