A mesh method for solving fourth-order quasilinear elliptic equations

M.M. Karchevsky

Kazan Federal University, Kazan, 420008 Russia
E-mail: Mikhail.Karchevsky@kpfu.ru
Received June 3, 2019

Full text pdf

DOI: 10.26907/2541-7746.2019.3.405-422

For citation: Karchevsky M.M. A mesh method for solving fourth-order quasilinear elliptic equations. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2019, vol. 161, no. 3, pp. 405–422. doi: 10.26907/2541-7746.2019.3.405-422. (In Russian)

Abstract

A mixed finite element method for solving the Dirichlet problem for a fourth-order quasilinear elliptic equation in divergent form was proposed and investigated. It was assumed that the domain in which the problem is solved is bounded and has a dimension greater or equal to two. When constructing the finite element scheme, all the second derivatives of the required solution were chosen as auxiliary unknowns. The usual triangulation of the domain by Lagrangian simplicial (triangular) elements of orders two and higher was used. Under the assumption that the operator of the original problem satisfies the standard conditions of bounded nonlinearity and coercivity, the existence of an approximate solution for any value of the discretization parameter was proved. The uniqueness of the approximate solution was established under tighter restrictions, namely, assuming the Lipschitz-continuity and the strong monotony of the differential operator. Under the same conditions, a two-layer iterative process was constructed, and the estimation of the convergence rate independent of the discretization parameter was proved. Accuracy estimates for the approximate solutions, optimal in the case of linearity of the differential equation, were obtained. The results of the application of the proposed technique to the problem of the equilibrium of a thin elastic plate were presented.

Keywords: mixed finite element method, accuracy estimates, iterative method, convergence rate estimates, theory of plates

Acknowledgments. The study was supported by the Russian Foundation for Basic Research (projects nos. 18-41-160014, 19-08-01184).

References

1. Mikhlin S.G. The Numerical Performance of Variational Methods. Groningen, Wolters-Noordhoff Publ., 1971. 373 p.

2. Sobolev S.L. Vvedenie v teoriyu kubaturnykh formul [Introduction to the Theory of Cubature Formulas]. Moscow, Nauka, 1974. 808 p. (In Russian)

3. Sobolev S.L., Vaskevich V.L. The Theory of Cubature Formulas. Springer, 1997. 415 p.

4. Mikhailov V.P. Existence of boundary values for metaharmonic functions. Sb. Math., 1999, vol. 190, no. 10, pp. 1417–1448. doi: 10.1070/SM1999v190n10ABEH000431.

5. Ciarlet P.G. Mathematical Elasticity. Vol. II: Theory of Plates. Amsterdam, North-Holland, 1997. 497 p.

6. Timoshenko S.P., Woinowsky-Krieger S. Theory of Plates and Shells. New York, McGraw-Hill, 1959. 595 p.

7. Happel J., Brenner H. Low Reynolds Number Hydrodynamics. Martinus Mijhoff Publ., 1983. 553 p.

8. Vorovich I.I. Nonlinear Theory of Shallow Shells. Springer, 1999. 388 p.

9. Ciarlet P.G. Mathematical Elasticity. Vol. III: Theory of shells. Amsterdam, North-Holland, 2000. 599 p.

10. Ciarlet P.G. The finite element method for elliptic problems. In: Classics in Applied Mathematics. SIAM, 2002. xxiii, 529 p.

 11.  Astrakhantsev G.P. A method for the approximate solution of the Dirichlet problem for the biharmonic equation. USSR Comput. Math. Math. Phys., 1977, vol. 17, no. 4, pp. 157–175. doi: 10.1016/0041-5553(77)90113-6.

12. Kobel'kov G.M. Iterative processes for some classes of difference schemes. Chislennye Metody Mekh. Sploshnoi Sredy, 1981, vol. 12, no. 6, pp. 38–48. (in Russian).

13. Kobel'kov G.M. Reduction of a boundary value problem for a biharmonic equation to a problem of Stokes type. Dokl. Akad. Nauk SSSR, 1985, vol. 283, no. 3, pp. 539–541. (In Russian)

14. Kobel'kov G.M. Reduction of a boundary value problem for the biharmonic equation to a problem with an operator of Stokes type. Sov. Math. (Izv. Vyssh. Uchebn. Zaved.), 1985, vol. 29, no. 10, pp. 39–59.

15. archevsky M.M. Some methods of solving the first boundary value problem for the biharmonic difference equation. USSR Comput. Math. Math. Phys., 1983, vol. 23, no. 5, pp. 41–47. doi: 10.1016/S0041-5553(83)80155-4.

16. Ryazhskikh V.I., Slyusarev M.I., Popov M.I. Numerical integration of a biharmonic equation in square field. Vestn. S.-Peterb. Univ. Ser. 10, 2013, no. 1, pp. 52–62. (In Russian)

17. Astrakhantsev G.P. On a mixed finite-element method in problems of shell theory. USSR. Comput. Math. Math. Phys., 1989, vol. 29, no. 5, pp. 167–176. doi: 10.1016/0041-5553(89)90195-X.

18. Dubinskii Yu.A. Quasilinear elliptic and parabolic equations of arbitrary order. Russ. Math. Surv., 1968, vol. 23, no. 1, pp. 45–91. doi: 10.1070/RM1968v023n01ABEH001233.

19. Vainberg M.M. Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. John Wiley & Sons, 1974. 356 p.

20. Bernardi C. Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal., 1989, vol. 26, no. 5, pp. 1212–1240.

21. Zlamal M. Curved elements in the finite element method. I. SIAM J. Numer. Anal., 1973, vol. 10, no. 1, pp. 229–240.

22. Korneev V.G. Skhemy metoda konechnykh elementov vysokikh poryadkov tochnosti [Schemes of High Orders of Accuracy for the Finite Element Method]. Leningrad, Leningr. Gos. Univ., 1977. 206 p. (In Russian)

23. Dautov R.Z., Karchevsky M.M. Vvedenie v teoriyu metoda konechnykh elementov [Introduction to the Theory of Finite Element Method]. Kazan, Kazan. Univ., 2011. 238 p. (In Russian)

24. Vainberg M.M. Variational Methods for the Study of Nonlinear Operators. San Francisco, London, Amsterdam, Holden-Day, 1964. 323 p.

25. Karchevsky M.M., Pavlova M.F. Uravneniya matematicheskoi fiziki. Dopolnitel'nye glavy [Equations of Mathematical Physics. Additional Chapters]. St. Petersburg, Lan', 2016. 276 p. (In Russian)

26. Samarsky A.A., Nikolaev E.S. Numerical Methods for Grid Equations. Vol. II: Iterative methods. Basel, Birkhauser. 1989. 502 p.

27. Karchevsky M.M. A mixed finite element method for nonlinear problems in the theory of plates. Russ. Math., 1992, vol. 36, no. 7, pp. 10–17.

28. Aubin J. P. Behavior of the error of the approximate solutions of boundary value problems for the linear elliptic operators by the Galerkin's and finite difference methods. Ann. Scuola Norm. Super. Pisa, 1967, vol. 21, pp. 599–637.

29. Strang G., Fix G.J. An Analysis of the Finite Element Method. Wellesley-Cambridge Press, 2008. 400 p.

30. Bahriawati C., Carstensen C. Three Matlab implementations of the lowest-order Raviart-Thomas Mfem with a posteriori error control. Comput. Methods Appl. Math., 2005, vol. 5, no. 4, pp. 333–361.

31. Dautov R.Z. Programmnaya realizatsiya metoda konechnykh elementov v MATLAB [Matlab Implementation of the Finite Element Method]. Kazan, Izd. Kazan. Univ., 2014. 108 p. (In Russian)

32. Demmel J.W. Applied Numerical Linear Algebra. SIAM, 1997. 419 p.

The content is available under the license Creative Commons Attribution 4.0 License.