Ключевое слово: «finite element method»

This paper examines fluid flow and transport modeling in porous media, focusing on the influence of Robin boundary conditions. A mixed finite element method is employed to approximate the Stokes equation governing fluid motion. Additionally, the transport equation for substance concentration is analyzed, with solutions obtained using the Streamline Upwind/Petrov-Galerkin (SUPG) method for different diffusion coefficients. The sequential solution approach integrates fluid dynamics and transport phenomena to enhance accuracy. Numerical results are presented for a three-dimensional model, highlighting the interplay between flow, transport, and boundary conditions. This study provides insights into the behavior of fluids in porous structures, aiding ap-plications in fields like hydrology, petroleum engineering, and environmental science.

The paper considers numerical modeling of thermoelasticity problems in heterogeneous areas. A modeling methodology based on the finite element method has been developed and implemented. The main problems of thermal conductivity, elasticity and thermoelasticity in homogeneous areas are investigated to verify the correctness of the methods. The analysis of the effect of the discretization of the calculated grid on the accuracy of solutions was carried out, which made it possible to determine the optimal grid parameters. Next, thermoelasticity modeling was performed in inhomogeneous areas, including materials with variable physical characteristics. The results of numerical experiments confirmed the effectiveness of the proposed methods, and also demonstrated the significant influence of heterogeneity on the temperature and mechanical charac-teristics of the system. The data obtained can be used to design structures subject to thermoelastic deformations in various engineering applications.

This article provides a concise review of the Continuous Galerkin (CG) and Discontinuous Galerkin (DG) methods, focusing on their application for numerical simulation of elastic wave propagation in heterogeneous media. The study is based on a comparative analysis of these two finite element approaches. The CG method, which enforces solution continuity across element boundaries, is recognized for its efficiency in simulations involving smooth media and homogeneous materials. In contrast, the DG method, which allows for solution discontinuities and uses numerical fluxes for inter-element coupling, demonstrates superior flexibility and accuracy when modeling complex geological structures. Such structures include media with sharp material contrasts, layered formations, and fracture networks. The DG formulation naturally yields a block-diagonal mass matrix, facilitating efficient explicit time integration and enhanced parallel scalability. The review also underscores the critical role of selecting appropriate iterative solvers and preconditioners, such as Krylov subspace methods combined with algebraic multigrid or incomplete LU factorization, to optimize computational performance. The practical application of both methods is illustrated through numerical experiments simulating wave propagation in layered and fractured media, confirming the DG method's robustness for problems with internal discontinuities.

The classical darcy model is insufficient for describing high-velocity filtration flows where inertial and viscous effects are significant. This paper presents a numerical comparison of the darcy, darcy-forchheimer, and the comprehensive darcy-forchheimer-brinkman models to quantify the impact of these phenomena. Using the finite element method with picard iteration, we simulate flow in a simple square domain. The results clearly show that the forchheimer inertial term introduces a non-linear pressure drop, while the brinkman viscous term smooths the velocity field by resolving viscous stresses. This study provides a clear numerical benchmark, highlighting the critical conditions under which each physical effect becomes dominant and underscoring the importance of model selection for accurate predictions in applications ranging from petroleum engineering to biomechanics.