Galerkin Finite Element Methods For Nonlinear Fractional Fourth-Order Partial Differential Equations

In this article, we use two-grid Galerkin mixed element method to solve two-dimensional nonlinear fourth-order reaction diffusion equation with Caputo time-fractional derivative and apply local discontinuous Galerkin (LDG) method to nonlinear a time-fractional fourth-order partial differential equation.In the first part, we apply two-grid mixed element method to two-dimensional nonlin-ear fourth-order reaction diffusion equation covering time-fractional derivative term. The integer derivative in time is approximated by second-order backward difference scheme and the time fractional derivative is discretized by the L1-approximation, and the spatial direction is discretized by mixed element combined with two-grid method. The pre-sented method mainly covers two computational steps:Solve a nonlinear system on the coarse grid, and then solve the linear system on the finer grid. The numerical theories on stability and error estimates are derived and the optimal space-time convergence rate O(kΔ2-α+hr+1+H2r+2) is obtained. Some numerical examples are shown to illustrate two-grid Galerkin mixed element method can improve the efficiency of computation.In the second part, LDG method is adopted to solve nonlinear time-fractional fourth-order partial differential equation. The integer-order derivative and fractional derivative in time are approximated by second-order backward difference scheme and WSGD-formula, the spatial derivative is discretized by LDG method. The detailed proof of the stability and error estimates is derived, and the second-order convergence rate with O(Δt2) is arrived at. Theoretical and numerical results show that the WSGD approximation can be well combined with LDG method.