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

L. L. Glazyrina   ; M. F. Pavlova

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 4, pp. 482-499

Résumé

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.

Détail

Keywords: double degeneration, non-local boundary conditions, method of semidiscretization, finite element method, generalized solution.

Bibliographie

[1] Antontsev S. N., Meirmanov A. M., Mathematical Models of Simultaneous Motions of Surface and Ground Waters, Izd. Novosib. Gos. Univ., Novosibirsk, 1979, 80 pp. (In Russian)

[2] Antontsev S. N., Epikhov G. R., Kashevarov A. A., Mathematical System Modelling of Water Exchange Processes, Nauka, Novosibirsk, 1986, 250 pp. (In Russian) | MR

[3] Timerbaev M. R., “Spaces with a graph norm and strengthened Sobolev spaces. I”, Russ. Math., 47:5 (2003), 52–62 | MR | Zbl

[4] Timerbaev M. R., “Spaces with a graph norm and stengthened Sobolev spaces. II”, Russ. Math., 47:9 (2003), 43–49 | MR | Zbl

[5] Glazyrina L. L., Pavlova M. F., “Convergence of implicit difference scheme for the problem of conjunctive ground water and surface flow with an arbitrary channel cross section”, J. Math. Sci., 71:6 (1994), 2744–2756 | DOI | MR | Zbl

[6] Glazyrina L. L., Pavlova M. F., “A uniqueness theorem for the solution of a problem in the theory of the joint motion of channel and underground waters”, Russ. Math., 44:11 (2000), 10–22 | MR | Zbl

[7] Glazyrina L. L., Pavlova M. F., “A uniqueness theorem for the solution of a problem in the theory of the joint motion of channel and underground waters with nonstationary boundary conditions”, Isseld. Prikl. Mat. Inf., 24, 2003, 31–42 (In Russian)

[8] Glazyrina L. L., Pavlova M. F., “On uniqueness of the solution of a variational inequality of the coupled movement of the underground and surface waters theory with nonhomogeneous bounds and nonhomogeneous boundary conditions”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 149, no. 4, 2007,, 73–89 (In Russian)

[9] Glazyrina L. L., Pavlova M. F., “On the solvability of the problem of the coupled movement of underground and surface waters with nonhomogeneous bounds to the solution”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 154, no. 1, 2012,, 147–161 (In Russian)

[10] Glazyrina L. L., Pavlova M. F., “Numerical solution of the nonstationary problem of conjunctive motion of ground and surface water”, J. Sov. Math., 61:6 (1992), 2417–2421 | DOI | MR

[11] Glazyrina L. L., Pavlova M. F., “The finite element method explicit scheme for a solution of one problem of surface and ground water combined movement”, IOP Conf. Ser.: Mater. Sci. Eng., 158:1 (2016.), Art. 012041, 7 pp.

[12] Kashevarov A. A., “Approximated Recording of the Vertical Flow in Hydraulic Filtration Models”, Boundary Problems of Filtration, Continuous Medium Dynamics, 108, Inst. Gidrodin. Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 1994, 3–13 (In Russian)

[13] Nobi N., Das Gupta A., “Simulation of regional flow and salinity intrusion in an integrated stream-aquifer system in coastal region: Southwest region of Bangladesh”, Ground Water, 35:5 (1997), 786–796 | DOI

[14] Glazyrina O. V., Pavlova M. F., “Study of the convergence of the finite-element method for solving parabolic equations with a nonlinear nonlocal space operator”, Differ. Equations, 51:7 (2015), 876–889 | DOI | MR | Zbl

[15] Glazyrina O. V., Pavlova M. F., “On the solvability of an evolution variational inequality with a nonlocal space operator”, Differ. Equations, 50:7 (2014), 873–887 | DOI | MR | Zbl

[16] Pavlova M. F., “On the solvability of nonlocal nonstationary problems with double degeneration”, Differ. Equations, 47:8 (2011), 1161–1175 | DOI | MR | Zbl

[17] Dautov R. Z., Karchevskii M. M., An Introduction to the Theory of Finite Element Methods, Kazan. Univ., Kazan, 2011, 240 pp. (In Russian)

[18] Korneev V. G., Schemes of the Method of Finite Elements for High Orders of Accuracy, LGU, Leningrad, 1977, 206 pp. (In Russian) | MR

[19] Lions J.-L., Some Methods of Solving Non-Linear Boundary Value Problems, Mir, Moscow, 1972, 588 pp. (In Russian) | MR

[20] Pavlova M. F., “Investigation of the equations of nonstationary nonlinear filtration”, Differ. Uravn., 23:7 (1987), 1436–1446 (In Russian) | MR | Zbl

[21] Alt H. W., Luckhaus S., “Quasilinear elliptic-parabolic differential equation”, Mat. Z., 183:8 (1983), 311–341 | MR | Zbl