Opuscula Math. 27, no. 2 (2007), 259-289

Opuscula Mathematica

# A finite difference method for nonlinear parabolic-elliptic systems of second order partial differential equations

Marian Malec
Lucjan Sapa

Abstract. This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in $$\mathbf{R}^{1+n}$$. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.

Keywords: partial differential equation, parabolic-elliptic system, finite difference method, finite difference scheme, consistence, convergence, stability, error estimate, uniqueness.

Mathematics Subject Classification: 65M06, 65M12, 65M15, 35M10, 39A10, 39B72.

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