# American Institute of Mathematical Sciences

February  2021, 14(2): 723-743. doi: 10.3934/dcdss.2020354

## Theoretical and numerical analysis of a class of quasilinear elliptic equations

 1 Université de Lorraine, CNRS, IECL, F 54000 Nancy, France 2 Laboratory LAMAI, Faculty of Science and Technology, University Cadi Ayyad, B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco 3 LAMMDA, Université de Sousse, ESST Hammam Sousse, Rue Lamine El Abbessi, Hammam Sousse, 4011, Tunisia

* Corresponding author: Nahed Naceur

Received  October 2019 Revised  December 2019 Published  February 2021 Early access  May 2020

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, $N\geq 1$, and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the non-negative weak solution of the considered equation has coupled the Newton method with domain decomposition and Yosida approximation of the nonlinearity. The domain decomposition is adapted to the nonlinearity at each step of the Newton method. Numerical examples are presented and commented on.

Citation: Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354
##### References:
 [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, D. Van Nostrand Co., Inc., Princeton, 1965. [2] N. Alaa, Etude d'Équations Elliptiques Non-Linéaires à Dépendance Convexe en le Gradient et à Données Mesures, Ph.D thesis, Université de Nancy I, 1989. [3] N. Alaa, A. Cheggour and J. R. Roche, Mathematical and numerical analysis of a class of non-linear elliptic equations in the two dimensional case, Numerical Mathematics and Advanced Applications, Springer, Berlin, 2006,926–934. doi: 10.1007/978-3-540-34288-5_92. [4] N. Alaa, F. Maach and I. Mounir, Existence for some quasilinear elliptic systems with critical growth nonlinearity and $L^{1}$ data, J. Appl. Anal., 11 (2005), 81-94.  doi: 10.1515/JAA.2005.81. [5] N. Alaa and I. Mounir, Global existence for reaction-diffusion systems with mass control and critical growth with respect to the gradient, J. Math. Anal. Appl., 253 (2001), 532-557.  doi: 10.1006/jmaa.2000.7163. [6] N. Alaa and M. Iguernane, Weak periodic solutions of some quasilinear parabolic equations with data measures, JIPAM. J. Inequal. Pure Appl. Math., 3 (2002), 14pp. [7] N. Alaa, Solutions faibles d'équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise Pascal, 3 (1996), 1-15.  doi: 10.5802/ambp.64. [8] N. Alaa and M. Pierre, Weak solution of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35.  doi: 10.1137/0524002. [9] N. Alaa and J. R. Roche, Theoretical and numerical analysis of a class of nonlinear elliptic equations, Mediterr. J. Math, 2 (2005), 327-344.  doi: 10.1007/s00009-005-0048-4. [10] H. Amann, Fixed point equations and nonlinear eigenvalue problems in order Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [11] H. Amann and M. G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.  doi: 10.1512/iumj.1978.27.27050. [12] P. Baras and M. Pierre, Critères d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 185-212.  doi: 10.1016/S0294-1449(16)30402-4. [13] A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 5 (1988), 347-364.  doi: 10.1016/S0294-1449(16)30342-0. [14] L. Boccardo, F. Murat and J. P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13 (1989), 373-392.  doi: 10.1016/0362-546X(89)90045-X. [15] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-75934-0. [16] H. Brezis and W. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565. [17] X. C. Cai and O. B. Widlund, Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems, SIAM J. Numer. Anal., 30 (1993), 936-952.  doi: 10.1137/0730049. [18] X. C. Cai and O. B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput., 13 (1992), 243-258.  doi: 10.1137/0913013. [19] X. C. Cai, An optimal two-level overlapping domain decomposition method for elliptic problems in two and three dimensions, SIAM J. Sci. Comput., 14 (1993), 239-247.  doi: 10.1137/0914014. [20] Y. Choquet-Bruhat and J. Leray, Sur le problème de Dirichlet, quasilineaire, d'ordre $2$, C. R. Acad. Sci. Paris, Sér. A-B, 274 (1972), A81–A85. [21] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New-York-Oxford, 1978. [22] P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, 35, Springer-Verlag, Berlin, 2004. [23] M. Dryja, B. F. Smith and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), 1662-1694.  doi: 10.1137/0731086. [24] M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap. Iterative methods in numerical linear algebra, SIAM J. Sci. Comput., 15 (1994), 604-620.  doi: 10.1137/0915040. [25] J. A. Ezquerro and M. A. Hernandez, On an application of Newton's method to nonlinear operators with $w$-conditioned second derivative, BIT, 42 (2002), 519-530.  doi: 10.1023/A:1021977126075. [26] J. A. Ezquerro and M. A. Hernandez, On the $mathbb{R}^{N}$-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math., 197 (2006), 53-61.  doi: 10.1016/j.cam.2005.10.023. [27] M. Gander, A waveform relaxation with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1999), 125-145.  doi: 10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4. [28] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0. [29] M. A. Hernandez, The Newton method for operators with Hölder continous first derivative, J. Optim. Theory Appl., 109 (2001), 631-648.  doi: 10.1023/A:1017571906739. [30] C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, 16, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944. [31] S. A. Levin, Models in ecotoxicology: Methodological aspects, in Applied Mathematical Ecology, Biomathematics, 18, Springer, Berlin, 1989,315–321. doi: 10.1007/978-3-642-61317-3_13. [32] P. L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal., 74 (1980), 335-353.  doi: 10.1007/BF00249679. [33] P. L. Lions, On the Schwarz alternating method. Ⅰ, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1988, 1–42. [34] J. D. Murray, Mathematical Biology, Biomathematics, 19, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [35] A. Porretta, Existence for elliptic equations in $L^{1}$ having lower order terms with natural growth, Portugal. Math., 57 (2000), 179-190. [36] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-540-85268-1. [37] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1999. [38] J. T. Schwartz, Nonlinear Functional Analysis, Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. [39] B. F. Smith, P. E. Bjorstad and W. D. Gropp, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.

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##### References:
 [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, D. Van Nostrand Co., Inc., Princeton, 1965. [2] N. Alaa, Etude d'Équations Elliptiques Non-Linéaires à Dépendance Convexe en le Gradient et à Données Mesures, Ph.D thesis, Université de Nancy I, 1989. [3] N. Alaa, A. Cheggour and J. R. Roche, Mathematical and numerical analysis of a class of non-linear elliptic equations in the two dimensional case, Numerical Mathematics and Advanced Applications, Springer, Berlin, 2006,926–934. doi: 10.1007/978-3-540-34288-5_92. [4] N. Alaa, F. Maach and I. Mounir, Existence for some quasilinear elliptic systems with critical growth nonlinearity and $L^{1}$ data, J. Appl. Anal., 11 (2005), 81-94.  doi: 10.1515/JAA.2005.81. [5] N. Alaa and I. Mounir, Global existence for reaction-diffusion systems with mass control and critical growth with respect to the gradient, J. Math. Anal. Appl., 253 (2001), 532-557.  doi: 10.1006/jmaa.2000.7163. [6] N. Alaa and M. Iguernane, Weak periodic solutions of some quasilinear parabolic equations with data measures, JIPAM. J. Inequal. Pure Appl. Math., 3 (2002), 14pp. [7] N. Alaa, Solutions faibles d'équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise Pascal, 3 (1996), 1-15.  doi: 10.5802/ambp.64. [8] N. Alaa and M. Pierre, Weak solution of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35.  doi: 10.1137/0524002. [9] N. Alaa and J. R. Roche, Theoretical and numerical analysis of a class of nonlinear elliptic equations, Mediterr. J. Math, 2 (2005), 327-344.  doi: 10.1007/s00009-005-0048-4. [10] H. Amann, Fixed point equations and nonlinear eigenvalue problems in order Banach spaces, SIAM Rev., 18 (1976), 620-709.  doi: 10.1137/1018114. [11] H. Amann and M. G. Crandall, On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J., 27 (1978), 779-790.  doi: 10.1512/iumj.1978.27.27050. [12] P. Baras and M. Pierre, Critères d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 185-212.  doi: 10.1016/S0294-1449(16)30402-4. [13] A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 5 (1988), 347-364.  doi: 10.1016/S0294-1449(16)30342-0. [14] L. Boccardo, F. Murat and J. P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13 (1989), 373-392.  doi: 10.1016/0362-546X(89)90045-X. [15] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 15, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-75934-0. [16] H. Brezis and W. Strauss, Semilinear second-order elliptic equations in $L^{1}$, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565. [17] X. C. Cai and O. B. Widlund, Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems, SIAM J. Numer. Anal., 30 (1993), 936-952.  doi: 10.1137/0730049. [18] X. C. Cai and O. B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput., 13 (1992), 243-258.  doi: 10.1137/0913013. [19] X. C. Cai, An optimal two-level overlapping domain decomposition method for elliptic problems in two and three dimensions, SIAM J. Sci. Comput., 14 (1993), 239-247.  doi: 10.1137/0914014. [20] Y. Choquet-Bruhat and J. Leray, Sur le problème de Dirichlet, quasilineaire, d'ordre $2$, C. R. Acad. Sci. Paris, Sér. A-B, 274 (1972), A81–A85. [21] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New-York-Oxford, 1978. [22] P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, 35, Springer-Verlag, Berlin, 2004. [23] M. Dryja, B. F. Smith and O. B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal., 31 (1994), 1662-1694.  doi: 10.1137/0731086. [24] M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap. Iterative methods in numerical linear algebra, SIAM J. Sci. Comput., 15 (1994), 604-620.  doi: 10.1137/0915040. [25] J. A. Ezquerro and M. A. Hernandez, On an application of Newton's method to nonlinear operators with $w$-conditioned second derivative, BIT, 42 (2002), 519-530.  doi: 10.1023/A:1021977126075. [26] J. A. Ezquerro and M. A. Hernandez, On the $mathbb{R}^{N}$-order of convergence of Newton's method under mild differentiability conditions, J. Comput. Appl. Math., 197 (2006), 53-61.  doi: 10.1016/j.cam.2005.10.023. [27] M. Gander, A waveform relaxation with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6 (1999), 125-145.  doi: 10.1002/(SICI)1099-1506(199903)6:2<125::AID-NLA152>3.0.CO;2-4. [28] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0. [29] M. A. Hernandez, The Newton method for operators with Hölder continous first derivative, J. Optim. Theory Appl., 109 (2001), 631-648.  doi: 10.1023/A:1017571906739. [30] C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, 16, SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944. [31] S. A. Levin, Models in ecotoxicology: Methodological aspects, in Applied Mathematical Ecology, Biomathematics, 18, Springer, Berlin, 1989,315–321. doi: 10.1007/978-3-642-61317-3_13. [32] P. L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Rational Mech. Anal., 74 (1980), 335-353.  doi: 10.1007/BF00249679. [33] P. L. Lions, On the Schwarz alternating method. Ⅰ, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1988, 1–42. [34] J. D. Murray, Mathematical Biology, Biomathematics, 19, Springer-Verlag, Berlin, 1993. doi: 10.1007/b98869. [35] A. Porretta, Existence for elliptic equations in $L^{1}$ having lower order terms with natural growth, Portugal. Math., 57 (2000), 179-190. [36] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, 23, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-540-85268-1. [37] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1999. [38] J. T. Schwartz, Nonlinear Functional Analysis, Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. [39] B. F. Smith, P. E. Bjorstad and W. D. Gropp, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996.
The computed super-solution ${{w}_h}$
The numerical approximation of the solution of (30)
Evolution of the error w.r.t. $n$
The numerical approximation of the super-solution of (31)
The final sub-domains decomposition
The computed solution at the 20th step of the algorithm implementing the Yosida's approximation of $G$
The $L^2$-norm of error between $u$ and $u_{n, h}$
 n 1 2 3 ... 8 9 10 Error $0.6$ $0.281$ 0.1033 ... $1.2 \; 10^{-4}$ $3.05\; 10^{-5}$ $8.21 \; 10^{-6}$ #Newton iteration 5 7 9 ... 7 6 6
 n 1 2 3 ... 8 9 10 Error $0.6$ $0.281$ 0.1033 ... $1.2 \; 10^{-4}$ $3.05\; 10^{-5}$ $8.21 \; 10^{-6}$ #Newton iteration 5 7 9 ... 7 6 6
The behavior of the algorithm computing the super-solution
 Newton iteration 1 2 3 4 Norm of the Newton update $4.21$ $0.17$ $0.026$ $6.4\; 10^{-4}$ # Schwarz iteration - 20 14 6 Norm of the Schwarz update - $8.6\; 10^{-4}$ $9.1\; 10^{-4}$ $8.8\; 10^{-4}$ #sub-domains 1 9 16 16
 Newton iteration 1 2 3 4 Norm of the Newton update $4.21$ $0.17$ $0.026$ $6.4\; 10^{-4}$ # Schwarz iteration - 20 14 6 Norm of the Schwarz update - $8.6\; 10^{-4}$ $9.1\; 10^{-4}$ $8.8\; 10^{-4}$ #sub-domains 1 9 16 16
The behavior of the algorithm computing the solution of the problem (20)
 n 1 2 ... 6 7 ... 19 20 Norm of the update 0.1009 0.1037 ... 0.0232 0.018 ... 0.0065 0.0062 # Newton iteration 7 8 ... 8 9 ... 9 8
 n 1 2 ... 6 7 ... 19 20 Norm of the update 0.1009 0.1037 ... 0.0232 0.018 ... 0.0065 0.0062 # Newton iteration 7 8 ... 8 9 ... 9 8
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