Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

Abstract

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.