# Construction of an iterative method for solving a nonlinear elliptic equation based on a mixed finite element method

• D.R. Baigereyev S. Amanzholov East Kazakhstan State University
• N.M. Temirbekov National Engineering Academy of the Republic of Kazakhstan
• D.A. Omariyeva D. Serikbayev East Kazakhstan State Technical University

### Abstract

This article is devoted to the construction and study of the finite element method for solving a two-dimensional nonlinear equation of elliptic type. Equations of this type arise in solving many applied problems, including problems of the theory of multiphase filtering, the theory of semiconductor devices, and many others. The relevance of the study of this problem is associated with the need to develop effective parallel methods for solving this problem. To discretize the equation, a mixed finite element method with Brezzi-Douglas-Marini elements is used. The issue of the convergence of the finite element method is investigated. To linearize the equation, the Picard iterative method is constructed. Two classes of basis functions of finite elements are used in the paper. A comparative analysis of the effectiveness of several direct and iterative methods for solving the resulting system of linear algebraic equations is carried out, including the method based on the Bunch-Kaufman LDLt factorization, the method of minimal residuals, the symmetric LQ method, the stabilized biconjugate gradient method, and a number of other iterative Krylov subspace algorithms with preconditioners based on incomplete LU decomposition. The method has been tested on several model problems by comparing an approximate solution with a known exact solution. The results of the analysis of the method error in various norms depending on the diameter of the mesh are presented.

### References

 Gatica G., Baier R. and Tierra, G. A mixed ﬁnite element method for Darcy’s equations with pressure dependent porosity // Mathematics of Computation. - 2015. - Vol. 297. - P. 1–33.
 Atkinson K. and Hansen O. Solving the Nonlinear Poisson Equation on the Unit Disk // Journal of Integral Equations.- 2005. - No. 3. - P. 223-251.
 Auricchio F., Veiga L., Brezzi F. and Lovadina C. Mixed ﬁnite element methods // Encyclopedia of Computational Mechanics Second Edition. - 2017. - No.1. - P. 1–53.
 Puscas M. A., Enchery G. and Desroziers, S. Application of the mixed multiscale ﬁnite element method to parallel simulations of two-phase ﬂows in porous media // Oil and Gas Science and Technology. - 2018. - Vol. 73, No. 38. - P. 1-14.
 Muzhinji K., Shateyi S. and Motsa S. The Mixed Finite Element Multigrid Method for Stokes Equations // The Scientiﬁc World Journal. - 2015. - No. 460421. - P. 1–12.
 Vorwerk J., Engwer C., Pursiainen S. and Wolters C. A mixed ﬁnite element method to solve the EEG forward problem // IEEE transations on medical imaging. - 2016. - No. 4. - P. 930–941.
 Spiridonov D., Huang J., Vasilyeva M., Huang Yu. and Chung, E. Mixed generalized multiscale ﬁnite element method for Darcy-Forchheimer model // Mathematics. - 2019. - Vol. 7, No. 1212. - P. 1–13.
 Brown D. L. and Vasilyeva M. Generalized multiscale ﬁnite element method for poroelasticity problems II: nonlinear coupling // Journal of Computational and Applied Mathematics. - 2016. - Vol. 297. - P. 132–146.
 Rebholz L., Viguerie A. and Xiao M. Eﬃcient nonlinear iteration schemes based on algebraic splitting for the incompress- ible Navier-Stokes equations // Mathematics of Computation. - 2019. - Vol. 88, No. 318. - P. 1533–1557.
 Islam M., Hye A. and Mamun A. Nonlinear Eﬀects on the Convergence of Picard and Newton Iteration Methods in the Numerical Solution of One-Dimensional Variably Saturated–Unsaturated Flow Problems // Hydrology. - 2017. - Vol. 4, No. 50. - P. 1–18.
 Kuraz M., Mayer P. and Pech P. Solving the nonlinear Richards equation model with adaptive domain decomposition // Journal of Computational and Applied Mathematics. - 2014. - Vol. 270. - P. 2-11.
 Nakshatrala K. and Turner D. A mixed formulation for a modiﬁcation to Darcy equation based on Picard linearization and numerical solutions to large-scale realistic problems // International Journal for Computational Methods in Engineering Science and Mechanics. - 2013. - Vol. 14, No. 6. - P. 524–541.
 Madzvamuse A. and Chung A. Fully implicit time-stepping schemes and non-linear solvers for systems of reaction–diﬀusion equations // Applied Mathematics and Computation. - 2014. - Vol. 244. - P. 361–374.
 Muccino J. and Luo H. Picard iterations for a ﬁnite element shallow water equation model // Ocean modeling. - 2005. - Vol. 10. - P. 316–341.
 Zha Y., Yang J., Yin L., Zhang Y., Zheng, W. and Shi L. A modiﬁed Picard iteration scheme for overcoming numerical diﬃculties of simulating inﬁltration into dry soil // Journal of Hydrology. - 2017. - Vol. 551. - P. 56–69.
 List F. and Radu F. A study on iterative methods for solving Richards’ equation // arXiv. - 2015. - Vol. 1507.07837v1. - P. 1–16.
 Baboulin M., Dongarra J., Remy A., Tomov S. and Yamazaki I. Solving dense symmetric indeﬁnite systems using GPU // Concurrency and Computation. - 2017. - Vol. 29, No. 9. - P. 1–17.
 Zhong-Zhi B. Motivations and realizations of Krylov subspace methods for large sparse linear systems // Journal of Computational and Applied Mathematics. - 2015. - Vol. 283, No. 1. - P. 71–78.
 Tran H., Toh K. and Phoon K. Preconditioned IDR(s) iterative solver for non-symmetric linear system associated with FEM analysis of shallow foundation // International Journal for Numerical and Analytical Methods in Geomechanics. - 2013. - Vol. 37. - P. 2972–2986.
 Choi S., Paige C. and Saunders M. MINRES-QLP: A Krylov subspace method for indeﬁnite or singular symmetric systems // SIAM Journal of Scientiﬁc Computing. - 2011. - Vol. 3, No. 4. - P. 1810–1836.
 Chronopoulos A. T. and Kucherov A. Block s-step Krylov iterative methods // Numerical Linear Algebra with Applica- tions. - 2010. - Vol. 17, No. 1. - P. 3–15.
 Cotter C. and Kirby R. Mixed ﬁnite elements for global tide models // Numerische Mathematik. - 2015. - No. 133. - P. 255–277.
 Chen Z. Finite element methods and their applications. - Springer, 2007. - 410 p.
 Kozulik, V. and Gotovac, B. Numerical solution of Poisson’s Equation in an arbitrary domain by using meshless R-function method // Proceeding of the 27th DAAM International Symposium on Intellect Manufacturing and Automation. - 2016.- P. 245–254.
Published
2020-06-26
How to Cite
BAIGEREYEV, D.R.; TEMIRBEKOV, N.M.; OMARIYEVA, D.A.. Construction of an iterative method for solving a nonlinear elliptic equation based on a mixed finite element method. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 106, n. 2, p. 104-120, june 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/766>. Date accessed: 23 sep. 2020. doi: https://doi.org/10.26577/JMMCS.2020.v106.i2.09.
Citation Formats
Section
Applied Mathematics
Keywords mixed finite element method, nonlinear Poisson equation, a priori estimate, iterative method, Brezzi-Douglas-Marini elements