The present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform norm C is given.

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Computational Methods in Applied Mathematics
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Matus, P., Gaspar, F., Hieu, L. M., & Tuyen, V. T. K. (2017). Monotone difference schemes for weakly coupled elliptic and parabolic systems. Computational Methods in Applied Mathematics, 17(2), 287–298. doi:10.1515/cmam-2016-0046