Finite Element Method And Numerical Simulation For Two Kinds Of Fractional Partial Differential Equations

In this thesis,the time fractional convection-diffusion equation and two-dimensional time fractional diffusion model with nonlinear term are numerically solved by finite ele-ment method.For the time fractional convection-diffusion equation,the time fractional derivative is approximated by using the approximation formula with order(3-?),the space direction is discretized by finite element method,and then a high-order is formulated to find the numerical solutions.To give the stability and error analysis,some useful lemmas are introduced and derived,and then the space-time convergence results covering O(hr+1 +?3-?)with the detailed process of proof are provided,which implies that the obtained time convergence order is higher than the order(2-?)arrived at by the general L1 formula.Finally,a numerical example with detailed numerical algorithms is provided to validate our theoretical results.For the nonlinear two-dimensional fractional diffusion model,the finite element scheme with second-order Crank-Nicolson approximation is given,the detailed numerical process is shown,and space-time convergence order and error results are listed.From the nu-merical results,one can see that space-time convergence rate is close to 2,which is in agreement with the theoretical results with linear element.