A numerical method for solving two-dimensional nonlinear parabolic problems based on a preconditioning operator
This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods.
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O. Axelsson and J. Karátson. Double sobolev gradient preconditioning for nonlinear elliptic problems. Numerical Methods for Partial Differential Equations, 23(5):1018–1036, 2007. https://doi.org/10.1002/num.20207
I. Faragó and J. Karátson. The gradient finite element method for elliptic problems. Computers & Mathematics with Applications, 42(8):1043–1053, 2001. https://doi.org/10.1016/S0898-1221(01)00220-6
I. Faragó and J. Karátson. Numerical Solution of Nonliner Elliptic Problems via Preconditioning Operators: theory and Applications. Advances in Computation, 11, Nova Science Publisher, New York, 2002.
I. Faragó, J. Karátson and S. Korotov. Discrete nonnegativity for nonlinear cooperative parabolic PDE systems with non-monotone coupling. Mathematics and Computers in Simulation, 139:37–53, 2017. https://doi.org/10.1016/j.matcom.2016.03.015
K. Höllig. Finite Element Methods with B-splines. Society for Industrial and Applied Mathematics, Philadelphia, 2003. https://doi.org/10.1137/1.9780898717532
K. Höllig and U. Reif. Nonuniform web-splines. Computer Aided Geometric Design, 20(5):277–294, 2003. https://doi.org/10.1016/S0167-8396(03)00045-1
K. Höllig, U. Reif and J. Wipper. Weighted extended B-spline approximation of Dirichlet problems. SIAM Journal on Numerical Analysis, 39(2):442–462, 2001. https://doi.org/10.1137/S0036142900373208
K. Höllig, U. Reif and J. Wipper. B-spline Approximation of Neumann Problems. Universitat Stuttgart, 2001-2.
J. Karátson and I. Faragó. Preconditioning operators and Sobolev gradients for nonlinear elliptic problems. Computers & Mathematics with Applications, 50(7):1077–1092, 2005. https://doi.org/10.1016/j.camwa.2005.08.010
V.V.K. Srinivas Kumar, B.V. Ratish Kumar and P.C. Das. Weighted extended B-spline method for the approximation of stationary Stokes problem. Journal of Computational and Applied Mathematics, 186(2):335–348, 2006. https://doi.org/10.1016/j.cam.2005.02.008
T. Kurics. Equivalent operator preconditioning for elliptic problems with nonhomogeneous mixed boundary conditions. Journal of Computational and Applied Mathematics, 235(2):437–449, 2010. https://doi.org/10.1016/j.cam.2010.05.047
A. Zakeri and A.H. Salehi Shayegan. Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems. Applied Mathematical Modelling, 38(2):775–783, 2014. https://doi.org/10.1016/j.apm.2013.06.018