Numerical solution of nonlinear differential boundary value problems using adaptive non-overlapping domain decomposition method

References

• Murray JD. Mathematical biology. Berlin: Springer; 1993. (Biomathematics; 18).
   Google Scholar
• Levin SA. Models in ecotoxicology: methodological aspects. In: Levin SA, Hallam ThG, Gross LJ, editors. Applied mathematical ecology. Vol. 18. Berlin: Springer; 1989. p. 315–321. (Biomathematics).
   Google Scholar
• Alaa N. Etude d'équations elliptiques non-linéaires à dépendance convexe en le gradient et à données mesures [dissertation]. Nancy: Université de Nancy I; 1989.
   Google Scholar
• Hani H, Khenissi M. On a finite difference scheme for blow up solutions for the Chipot–Weissler equation. Appl Math Comput. 2015;268:1199–1216.
   Web of Science ®Google Scholar
• Alaa N, Roche JR. Theoretical and numerical analysis of a class of nonlinear elliptic equations. Mediterr J Math. 2005;2:327–344. doi: https://doi.org/10.1007/s00009-005-0048-4
   Google Scholar
• Lions PL. Résolution de problèmes elliptiques quasilinéaires. Arch Rational Mech Anl. 1980;74:335–353. doi: https://doi.org/10.1007/BF00249679
   Google Scholar
• Bensoussan A, Boccardo L, Murat F. On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann Inst Henri Poincaré. 1989;5:347–364. doi: https://doi.org/10.1016/S0294-1449(16)30342-0
   Google Scholar
• H. Amann H, Crandall MG. On some existence theorems for semilinear elliptic equations. Indiana Univ Math J. 1978;7:779–790. doi: https://doi.org/10.1512/iumj.1978.27.27050
   Google Scholar
• Boccardo L, Murat F, Puel JP. Existence results for some quasilinear parabolic equations. Nonlinear Anal Theory Method Appl. 1989;13:373–392. doi: https://doi.org/10.1016/0362-546X(89)90045-X
   Google Scholar
• Baras P, Pierre M. Critéres d'existence de solutions positives pour des equations semilineaires. Ann Fourier Grenoble. 1984;24:1985–2006.
   Google Scholar
• Alaa N, Maach F, Mounir I. Existence for some quasilinear elliptic systems with critical growth nonlinearity and L1 data. J Appl Anal. 2005;11:81–94. doi: https://doi.org/10.1515/JAA.2005.81
   Google Scholar
• Alaa N, Mounir I. Global existence for reaction–diffusion systems with mass control and critical growth with respect to the gradient. J Math Anal Appl. 2001;253:532–557. doi: https://doi.org/10.1006/jmaa.2000.7163
   Google Scholar
• Lions PL. On the Schwarz alternating method I. In: Chan TF, Glowinski R, Périaux J, Widlund O, editors. First international symposium on domain decomposition methods for partial differential equations. Philadelphia: SIAM; 1989.
   Google Scholar
• Lions PL. On the Schwarz alternating method III: a variant for non-overlapping subdomains. In Chan TF, Glowinski R, Périaux J, Widlund O, editors. Third international symposium on domain decomposition methods for partial differential equations. Philadelphia: SIAM; 1990.
   Google Scholar
• Smith BF, Bjorstad P, Gropp WD. Domain decomposition: parallel multilevel algorithms for elliptic partial differential equations. Cambridge: Cambridge University Press; 1996.
   Google Scholar
• A. Quarteroni A, Valli A. Domain decomposition methods for partial differential equations. Oxford: Oxford Science Publications; 1999.
   Google Scholar
• Mathew Tarek PA. Domain decomposition methods for the numerical solution of partial differential equations. Berlin: Springer; 2008. (Lecture Notes in Computational Science and Engineering; 61).
   Google Scholar
• Zou J. A new fast solver–monotone MG method (MMG). J Comput Math. 1987;5:325–335.
   Web of Science ®Google Scholar
• Zou J, Huang H-C. Algebraic subproblem decomposition methods and parallel algorithms with monotone convergence. J Comput Math. 1992;10:47–59.
   Google Scholar
• Hoffmann K-H, Zou J. Parallel algorithms of Schwarz variant for the variational inequalities. Numer Funct Anal Optim. 1992;13:449–462. doi: https://doi.org/10.1080/01630569208816491
   Web of Science ®Google Scholar
• Hoffmann K-H, Zou J. Parallel solution of varational inequality problems with nonlinear source terms. IMA J Numer Funct Anal. 1992;13(1):31–45.
   Google Scholar
• Badea L. On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems. SIAM J Numer Anal. 1991;28:179–204. doi: https://doi.org/10.1137/0728010
   Google Scholar
• Cai XC, Widlund OB. Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems. SIAM J Numer Anal. 1993;30:936–952. doi: https://doi.org/10.1137/0730049
   Web of Science ®Google Scholar
• Cai XC, Widlund OB. Domain decomposition algorithms for indefinite elliptic problems. SIAM J Sci Stat Comput. 1992;13:243–258. doi: https://doi.org/10.1137/0913013
   Web of Science ®Google Scholar
• Cai XC. An optimal two-level overlapping domain decomposition method for elliptic problems in two and three dimensions. SIAM J Sci Comput. 1993;14:239–247. doi: https://doi.org/10.1137/0914014
   Web of Science ®Google Scholar
• Pao CV. Block monotone iterative methods for numerical solutions of nonlinear elliptic equations. Numer Math. 1995;72:239–262. doi: https://doi.org/10.1007/s002110050168
   Web of Science ®Google Scholar
• Lui SH. On monotone and Schwarz alternating methods for nonlinear elliptic PDEs. ESAIM: Math Modell Numer Anal. 2001;35(1):1–15. doi: https://doi.org/10.1051/m2an:2001104
   Google Scholar
• Baras P. Semilinear problem with convex nonlinearity. In: Benilan Ph, Chipot M, Evans LC, Pierre M, editors. Recent advances in nonlinear elliptic and parabolic problems. Harlow: Longman Sci. Tech.; 1989. p. 201–215. (Pitman Research Notes in Mathematics Series; 208).
   Google Scholar
• Agmon S. Lectures on elliptic boundary value problems. Princeton: D. Van Nostrand Company; 1965.
   Google Scholar
• Deuflhard P. Newton methods for nonlinear problems. Berlin: Springer; 2000; (Springer Series in Computational Mathematics).
   Google Scholar
• Hernandez MA. The Newton method for operators with Hölder continuous first derivative. J Optim Theory Appl. 2001;109:631–648. doi: https://doi.org/10.1023/A:1017571906739
   Web of Science ®Google Scholar
• Ascher M, Mattheij RM, Russell RD. Numerical solution of boundary value problem of ordinary differential equations. Philadelphia: SIAM; 1995.
   Google Scholar
• Ascher UM, Russell RD. Reformulation of boundary value problems into ‘standart’ form. SIAM Rev. 1981;23:238–254. doi: https://doi.org/10.1137/1023039
   Google Scholar
• Bader G, Ascher U. A new basis implementation for a mixed order boundary value ODE solver. SIAM J Sci Stat Comput. 1987;8:483–500. doi: https://doi.org/10.1137/0908047
   Web of Science ®Google Scholar
• Ascher U. Collocation for two-point boundary value problems revisited. SIAM J Numer Anal. 1986;23:596–609. doi: https://doi.org/10.1137/0723038
   Google Scholar
• Kierzenka J, Shampine LF. A BVP solver based on residual control and the MATLAB PSE. ACM Trans Math Softw. 2001;27:299–316. doi: https://doi.org/10.1145/502800.502801
   Web of Science ®Google Scholar
• Shampine LF, Gladwell I, Thompson S. Solving ODE's with MATLAB. Cambridge: Cambridge University Press; 2003.
   Google Scholar
• Shampine LF, Reichelt MW, Kierzenka J. A solving boundary value problem for ordinary differential equation in MATLAB with BVP4C. Philadelphia: SIAM; 2000. p. 163–169.
   Google Scholar
• Ghosh P. Numerical, symbolic and statistical computing for chemical engineers using MATLAB. Delhi: PHI Learning Private Limited; 2018.
   Google Scholar
• Mazzia F, Cash JR, Soetaert K. Solving boundary value problems in the open source software R: package bvpSolve. Opuscula Math. 2014;34:387–403. doi: https://doi.org/10.7494/OpMath.2014.34.2.387
   Google Scholar
• Cash JR, Hollevoet D, Mazzia F, Nagy M. Algorithm 927: the MATLAB code bvptwp.m for the numerical solution of two boundary value problems. ACM Trans Math Softw. 2013;39: 12pp. doi: https://doi.org/10.1145/2427023.2427032
   Google Scholar