Algebraic aspect of the discrete maximum principle

Extension of the algebraic aspect of the discrete maximum principle

“…In conclusion, it is appropriate to consider the particular case where the diagonal elements of the matrix A 0 are positive and the other elements of the whole matrix A are nonpositive [9]. In this case we may add two more theorems to three presented above.…”

“…In this case we may add two more theorems to three presented above. Below we present only the formulations of these theorems, their proofs are given in detail in [9]. …”

“…This paper is the logical continuation and extension of the topic presented in [9]. The discrete maximum principle duplicating the principle with the same name from the mathematical physics is well known in numerical mathematics [4,7,8]; it is usually presented in several variants (relative to particular difference schemes) or in a more general form oriented to difference algorithms with an arbitrary stencil [1, 6 -8].…”

“…In this paper (as well as in [9]) an extended maximum principle is presented in the formal language of linear algebra without using any notions from the theory of difference schemes (difference equations, grid boundary and initial conditions, difference grid stencils, a grid node neighborhood, etc.). In particular, one may hypothetically consider a system of equations with a dense matrix, i.e., a matrix without any zero elements.…”

On discrete maximum principles for nonlinear elliptic problems

“…Other works devoted to various aspects of DMPs and related issues include [1], [8], [13], [21], [22], [31], [32], [33], and [34]. Several examples of real-life problems for which the validity of DMP is essential are given, e.g., in papers [3], [14], [25], and [37].…”