Optimal error estimates of mixed finite element methods for SPDEs with gradient-dependent multiplicative noise

Abstract This paper develops and analyses the mixed finite element methods (MFEMs) for Itô-type stochastic partial differential equations (SPDEs) driven by gradient-dependent multiplicative noise. Solutions to such SPDEs may lose regularity rapidly or potentially blow up in finite time due to gradient-dependent stochastic effects. By treating the gradient of the solution as an independent variable in MFEM, we explicitly track the influence of the gradient $\nabla u$ on the proposed schemes by setting the specified coefficient parameters relationship in the proof. We rigorously prove that the proposed semidiscrete and fully discrete schemes can theoretically achieve optimal strong convergence rates in space and time. Numerical tests are also presented to validate the theoretical results and demonstrate the effectiveness of our proposed method for SPDEs with gradient-dependent stochasticity.