Energy estimate, optimal solution and convergence analysis for interfaced valued schemes of the fluid–solid interaction

This study investigates the mathematical and numerical properties of energy estimate, regularity, and convergence analysis through the finite element method FEM for coupled bi-domain scenarios with two interface conditions. This includes the Dirichlet-Neumann (DN) coupling, which ensures continuity in flow problems, and the heat flux condition, which governs heat transfer between two regions. These results ensure that solutions are well defined, smooth and bounded, with energy behavior properly considered, which supports the reliability of numerical simulations. The results demonstrate that the DN-coupling conserves energy using one-sided finite differences without imposing stability constraints. In contrast, the heat flux condition, discretized with central differences, requires additional stability measures. Numerical simulations validate the theoretical findings, and the results are presented through detailed graphs and tables. This work provides a solid foundation both analysis and numerical point of view for exploring higher-order approximations and extending the model to two- and three-dimensional problems in complex multi-physics systems.