<p>This paper presents a robust and efficient numerical framework for solving a nonlinear distributed-order space–time fractional partial differential equation in one and higher dimensions. For spatial discretization, the finite element method is developed by employing nodal basis functions. Temporal discretization is achieved employing the second-order backward differentiation formula, combined with the composite trapezoidal rule and the weighted and shifted Grünwald–Letnikov approximation. The stability and convergence of the proposed numerical schemes are thoroughly analyzed using the energy method, ensuring their reliability. In order to verify the accuracy and effectiveness of the scheme, a series of numerical experiments are conducted, validating the theoretical findings.</p>