GENERALIZED SOLUTIONS AND NUMERICAL METHODS FOR STUDYING BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC EQUATIONS IN MULTIDIMENSIONAL DOMAINS

Аннотация

This scientific article presents an expanded, fundamental, deep, and comprehensive mathematical analysis of the existence, uniqueness, structural stability, and regularity of generalized solutions to second-order nonlinear elliptic equations in bounded multidimensional domains with smooth boundaries. The study examines in detail the functional-theoretical and topological foundations of variational inequality methods, topological degree theory, fixed-point theorems in Sobolev spaces, and the fundamental properties of monotone, pseudomonotone, and coercive operator mappings. Special attention is paid to the rigorous decomposition and structural analysis of discrete analogs of boundary value problems obtained through the application of the finite element method (FEM), adaptive mesh refinement procedures, and projection-grid methods. A comprehensive comparative analysis regarding the convergence rates of projection schemes, the spectral stability of numerical algorithms, and the local approximation properties of higher-order basis functions is carried out. Furthermore, the paper formulates and strictly proves advanced theorems concerning the asymptotic convergence rates of approximate finite element solutions toward the exact generalized solution in both natural energy norms and Lp Lebesgue spaces. The practical value of the obtained mathematical results lies in the possibility of directly applying the developed algorithms and theoretical frameworks to the computer simulation and numerical modeling of complex nonlinear physical processes, including non-Newtonian hydrodynamics, nonlinear elasticity theory, anisotropic heat conduction, and electrostatics in heterogeneous media.