Semidiscrete Finite Volume Element Methods For Semilinear Parabolic Integro-Differential Equations

The paper discuss the semidiscrete finite volume element methods for semilinear parabolic integro-differential equations.In the first place,the paper introduce weak formulation of semilinear parabolic integro-differential equations,then construct the triangulation of the spatial,formulate linear finite element space,and build finite volume element approximation.In the meantime,the paper also introduce Ritz - Volterra projections,and obtain corresponding estimates of error. Subsequently,the paper prove a sequence of lemmas,and obtain optimal-order L²-error estimate for smooth initial data.For nonsmooth initial data,having the aid of the backward problem of original equation,the paper derive an error estimate of order O(t^-1h² ln h) in the L²-norm.It is almost optimal.