A Dissipation‐Preserving Finite Element Approach for Nonlinear Nonlocal Euler–Bernoulli Equations

This paper focuses on the numerical solution of the modified Euler–Bernoulli equation incorporating nonlinear nonlocal viscous damping, which is suitable for modeling vibration phenomena in beams and plates. This formulation is relevant for modeling vibrational phenomena in engineering structures. The proposed strategy produces an approximation of the unknown solution through a two‐step numerical process. First, the Galerkin finite element method is used to derive a spatially semi‐discrete scheme. Then, a time‐stepping difference method is applied to accomplish a fully discrete Galerkin scheme. The energy method is employed to establish the stability and uniqueness of both the semi‐discrete and fully discrete solutions. The main difficulty in this work lies in handling the nonlinear viscous damping term during discretization and analysis. Optimal error estimates are obtained for both discretizations. Numerical simulations confirm the theoretical predictions and indicate that the fully discrete scheme efficiently dissipates energy.