Finite Volume Element Methods For Several Kinds Of Fractional Partial Differential Equations

The main work of this thesis is to study the finite volume element(FVE)methods for several kinds of time fractional partial differential equations(FPDEs),and propose a fast time two-mesh FVE methods for two kinds of nonlinear FPDEs by combining the idea of the time two-mesh method with the FVE method.For several kinds of Caputo type and Riemann Liouville type time FPDEs,the fully discrete numerical schemes are established,the theoretical analysis results of the stability and convergence of the fully discrete schemes are given,and numerical experiments are carried out to verify the the-oretical results by the experimental data.In this thesis,the fractional reaction-diffusion equation,the nonlinear fractional mobile/immobile transport equation,the nonlinear frac-tional fourth-order reaction-diffusion equation,the nonlinear fractional Cable equation,and the nonlinear fractional coupled diffusion system are considered.The detailed re-search contents can be summarized as the following three parts:In Chapter 2,the FVE method for a class of Caputo type time fractional reaction-diffusion equations is studied.In the time direction,the classical L1 formula is used to approximate the Caputo type time fractional derivative.In the process of the spatial discretization,primal and dual partitions are constructed for a two-dimensional bounded convex polygon domain,the piecewise linear polynomial function space and piecewise con-stant function space are selected as trial function space and test function space,respec-tively.Then,a fully discrete FVE scheme is constructed by introducing an interpolation operator Ih*.Making use of the properties of the L1 formula and the interpolation opera-tor Ih*,the existence,uniqueness,unconditional stability in L2 norm,and the conditional stability in H1 norm of the fully discrete solution are given,and optimal a priori error estimates are obtained.Finally,two numerical examples with different spatial dimensions are given to verify the feasibility of the numerical method.In Chapter 3 and Chapter 4,the FVE method for nonlinear time fractional mo-bile/immobile transport equations and the mixed finite volume element(MFVE)method for nonlinear time fractional fourth-order reaction-diffusion equations are studied.The second-order WSGD formula and a second-order linearized formula are used to approxi-mate the Riemann-Liouville time fractional derivative and the nonlinear term of two kinds of equations,respectively.When dealing with the nonlinear time fractional fourth-order reaction-diffusion equation,the primal problem is transformed into a low-order coupled system by introducing an auxiliary variable.Thus,by introducing the interpolation op-erator Ih*,the second-order fully discrete FVE scheme and MFVE scheme are established for two kinds of equations.Making use of the properties of the WSGD formula and in-terpolation operator Ih*,the existence,uniqueness and unconditional stability of the fully discrete scheme solutions are derived,and optimal a priori error estimates are obtained,in which the convergence orders are independent of the fractional parameters.Finally,numerical examples with different nonlinear terms for these two kinds of equations are given to verify our theoretical analysis resultsIn Chapter 5 and Chapter 6,fast time two-mesh FVE methods for the nonlinear frac-tional Cable equation and nonlinear fractional coupled diffusion systems with Riemann-Liouville time fractional derivative are studied.The calculation process of the time two-mesh FVE method is divided into three steps:firstly,the coarse solution is computed iteratively by using the nonlinear FVE scheme on the time coarse mesh;further,the coarse solution on the time fine mesh is calculated by using the Lagrange interpolation formula and the coarse solutions on the time coarse mesh:finally,a linearized FVE scheme is constructed by using the coarse solution on the time fine mesh and a special technique,and the time two-mesh solution on the time fine grid is obtained.This method can greatly reduce the calculation time and improve the calculation efficiency.The time two-mesh FVE schemes for these two equations are constructed,the unconditional stability and op-timal a priori error estimates are obtained for both coarse and fine time mesh.Finally,the corresponding numerical examples are given for these two equations.By comparing the numerical results of the time two-mesh method and the general FVE method,it can be seen that the time two-mesh FVE method can not only ensure the convergence accuracy,but also save the calculation time.