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In this paper, a stabilized Stokes–Stokes system with Nitsche's type interface conditions is presented. These conditions are commonly employed in many multi-physical fields, including fluid–fluid interaction, fluid–structure interaction, oceanographic modeling, and atmospheric forecasting. For multi-physical domain modeling purposes, Nitsche's interface conditions provide useful benefits over classical conditions via addressing the complicated nature of fluid phase interface mathematical modeling, phase boundary tracking, interface interactions, and mass and energy transportation. It is not easy to find analytical and numerical solutions for models with these characteristics. We use more accurate interface conditions to solve the fluid–fluid interaction model to accomplish this numerically. This is achieved by including new terms at the interface and decoupling the domain through the two-grid technique, which ultimately reduces the main issue into several smaller problems. Comparing this method to existing models, we find that it is computationally feasible because it uses less memory and operates with a coarse grid instead of a fine grid and thus improves convergence rates for complex and nonlinear problems. Furthermore, it shows mesh independence, supports potential parallelization, and is crucial for advanced multigrid techniques. The optimality of the error is confirmed both theoretically and numerically. The numerical experimental section validates the model through three types of numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity b ( x ) , and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are P 1 -discontinuous, P 2 -continuous and P 1 -continuous finite elements respectively. [ABSTRACT FROM AUTHOR]