Finite Element Method For Two-dimensional Fourth-order Nonlinear Modified Time-fractional Diffusion Equation

In this thesis,we consider the finite element method for two-dimensional nonlinear modified Riemann-Liouville time-factional fourth-order diffusion equation.Because of the fourth-order space derivative,in order to avoid using higher order elements,we intro-duce an intermediate variable ? = ?u and reduce the original fourth-order problem into a second-order coupled system.Based on the relation between Riemann-Liouville fractional derivative and Caputo fractional derivative,we discretize the Riemann-Liouville fractional time derivative terms by using the Ll-approximation and approximate the time direction by using the second-order backward difference formula,and approximate the space direc-tion by making use of finite element method.In chapter 1,we introduce developments for the numerical methods of fractional partial differential equations and provide a studied problem;In chapter 2,we formulate the fully discrete scheme with second-order backward difference scheme and an approximate formula for fractional derivative;In chapter 3,we derive a stable inequality in detail;In chapter 4,we prove the theoretical error results,from which one can see that the space convergence rate is optimal and the same convergence order with O(?t min {1+?,1+?)to the results obtained by L1 formula;In chapter 5,we make the numerical tests by choosing two numerical examples including a one-dimensional case for verifying the time convergence results and a two-dimensional case for the effectiveness in high-dimensional cases,which implies that numerical convergence rate is in agreement with the theoretical results.