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.

Maximum Principle, Monotone Schemes, Non-Uniform Grids, Two-Side Estimates, Uniform Norm, Weakly Coupled Elliptic System
Computational Methods in Applied Mathematics
This work was funded by the European Commission 7th Framework Programme; grant id h2020/705402 - Efficient numerical methods for deformable porous media. Application to carbon dioxide storage. (poro sos)
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Matus, P, Gaspar, F.J, 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