Application of the Method of Two-Sided Approximations Based on the Use of the Green – Rvachev Quasi-Function for the Numerical Analysis of Thermochemical Processes

Mathematical modelling of thermochemical processes plays a crucial role in the study and prediction of complex heat transfer phenomena and chemical reactions. In this field, a primary scientific challenge is the development of advanced numerical methods aimed at improving computational accuracyand expanding the practical applicability of mathematical models. Of particular importance is the design of numerical methods that not only provideapproximate solutions but also ensure convenient and reliable error estimation. This paper considers both two-dimensional and three-dimensionalDirichlet boundary value problems for a second-order elliptic equation with an exponentially nonlinear right-hand side. The problem under studyserves as a mathematical model of a thermochemical process governed by Arrhenius kinetics. For the numerical analysis, the method of two-sided approximations is proposed, based on the use of the Green – Rvachev quasi-function constructed within the framework of the theory of R - functions.Within this approach, the original boundary value problem is reduced to an equivalent Urysohn integral equation, which is analyzed using methods ofnonlinear analysis in semi-ordered Banach spaces. The properties of the corresponding integral operator, including heterotonicity, continuity, andcomplete continuity, are investigated. The construction of a strongly invariant conical segment for the heterotone operator is studied, and an iterative two-sided approximation scheme is proposed along with conditions ensuring its convergence. The performance of the proposed method is illustrated by computational experiments in the unit square and unit cube for test parameter values of the mathematical model. The obtained results are analyzed, and practical recommendations regarding the applicability of the developed methods are provided. Among the main advantages of the two-sided iterative schemes are the availability of a guaranteed a posteriori error estimate and a convenient stopping criterion for the computational process.