NUMERICAL METHODS FOR COUPLED SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

C. V. Pao; Taiping He

doi:10.11948/2016039

2016 Volume 6 Issue 2

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C. V. Pao, Taiping He. NUMERICAL METHODS FOR COUPLED SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 543-581. doi: 10.11948/2016039

Citation:

C. V. Pao, Taiping He. NUMERICAL METHODS FOR COUPLED SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2016, 6(2): 543-581. doi: 10.11948/2016039

NUMERICAL METHODS FOR COUPLED SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

C. V. Pao¹,
Taiping He²

1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA.;
2 SAS Institute, Inc, 100 SAS Campus Drive, Cary, NC 27513, USA

Abstract

Abstract

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution.
- Quasilinear elliptic equations /
- degenerate diffusion /
- monotone iterative schemes /
- upper and lower solutions /
- convergence of discrete solution /
- Lotka-Volterra cooperation system
MSC: 65N22;65N06;65N12

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