Optimal H~1-norm Error Estimates Of Several Fast Schemes For Nonlinear Subdiffusion Equations

The nonlinear subdiffusion equation has three important features:(1)The solution has initial singularity;(2)The fractional derivative is nonlocal;(3)The problem is nonlinear.These features bring some difficulties to numerical solution and theoretical analysis.Firstly,due to the initial singularity of the solution,if the usual uniform time mesh is used,the overall convergence order of the numerical scheme is relatively low.In order to obtain the uniform optimal convergence order,the non-uniform time mesh is generally used to overcome the singularity of the initial value,which also brings challenges to the numerical theoretical analysis.Secondly,because the fractional derivative is a nonlocal operator,the numerical simulation of long-time or small step size requires huge computational effort.In order to overcome the problem of huge computing storage and cost,a fast algorithm based on sum-of-exponential(SOE)approximation is adopted in this paper.Finally,for the nonlinear term,we use the explicit or linearized numerical scheme to avoid the problem that the implicit scheme needs numerical iteration and brings additional computation.The dissipative term is treated implicitly,and the spatial direction is approximated by finite element method,so an explicit implicit finite element numerical scheme based on fast algorithm is obtained.An important consideration of the explicit implicit scheme is whether the scheme is unconditionally stable and convergent,that is,whether the constraints between time and space meshes are required.Based on the above considerations,this paper mainly discusses the following three aspects:(1)For Caputo fractional derivative,L1 numerical discretization based on SOE fast algorithm is adopted,the dissipation term is discretized implicitly,and the nonlinear term is discretized by Newton linearization,the spatial direction is discretized by Galerkin finite element method,so a fast Ll-Galerkin finite element fully discrete numerical scheme is obtained.By introducing the following technologies and concepts:discrete complementary convolution(DCC)kernel,error convolution structure(ECS),discrete fractional Gr?nwall inequality and spacetime splitting method,we obtain the unconditionally optimal H1-norm error estimate of the Newton linearized fully discrete scheme with fast L1 approximation,and verify the correctness of the theoretical analysis through numerical simulations.(2)For linear subdiffusion equation,we first reformulate Caputo fractional derivative into RiemannLiouville fractional derivative,and then use L1R scheme to approximate the Riemann-Liouville fractional derivative.The corresponding discrete fractional Gr on wall inequality is proved.Using the newly established discrete fractional Gronwall inequality,the space-time splitting method and the appropriate test function,we further obtain the unconditionally optimal H1-norm error estimation of L1R approximation,Newton linearization and Galerkin finite element discrete scheme for the nonlinear subdiffusion equation,and the correctness of the theoretical analysis is verified by numerical simulations.At the same time,in order to speed up the numerical evaluation of Riemann-Liouville fractional derivative,a SOE fast LlR scheme is presented.The corresponding numerical theory is established,and the unconditionally optimal H1-norm error estimate of the fast L1R finite element fully discrete scheme is obtained,and the correctness of the theoretical analysis is verified by numerical simulations.(3)For Caputo fractional derivative,a higher order L2-1σ scheme with fast algorithm is considered by combining Galerkin finite element method in space.The nonlinear term is discretized by the Newton linearized method.Using the discrete fractional Gr?nwall inequality and spacetime splitting method,the unconditionally optimal H1-norm error estimate of our proposed scheme is obtained.Finally,the correctness of the theoretical analysis is verified by numerical simulations.