Siryk S. Computational schemes and weight functions of finite element Petrov-Galerkin method for high-precision modeling of convection-diffusion processes

The thesis is devoted to developing efficient computational schemes and weight functions of finite-element Petrov-Galerkin method to increase the accuracy, speed up the convergence of computer modeling of convection-diffusion processes, increase stability of simulation results and accelerate its computer implementation. For modelling non-stationary processes of convection-diffusion by Petrov-Galerkin method the new computational schemes in the form of ordinary differential equations and one-step difference schemes with weights have been developed as well as proved theoretically that the order of its maximum attainable local approximation accu-racy is two units higher than corresponding order of SUPG, GLS, USFEM methods and Petrov-Galerkin method with piecewise quadratic weighted functions. A new approach for improving the accuracy of modeling convection-diffusion processes by Petrov-Galerkin methods with us-ing mass lumping has been proposed. The software was developed to implement the proposed computational schemes and have their practical aplications for research in the theory of heat and mass transfer, hydrodynamics, magnetohydrodynamics, plasma physics. Experimental re-search and practical applications of the proposed weighting functions, numerical schemes and methods have shown an increase in the accuracy of modeling for various applications in the 1.2-4.5 times.