Research On Decoupling Algorithm For Parabolic Optimal Control Problems

Optimal control problem（OCP）of Partial differential equation（PDE）covers a wide range of fields.It has been applied in industrial development,transportation,economic management,fluid dynamics and other fields.Because of the complexity of the problem model in practical application,it is very difficult to solve the exact solution,so the numerical solution of the problem has important theoretical significance and practical value.This thesis studies the numerical solution of parabolic OCP using Multipoint flux mixed finite element（MFMFE）.Semi-discretization and full discretization of state variables of parabolic OCP using MFMFE method,with special quadrative formulas that allow local flux elimination,can decouple state and co-state variables into element-centered difference schemes instead of solving saddle point type algebraic systems.The error estimates and convergence orders of the two discrete schemes are derived respectively.When using MFMFE method to study the semi-discrete scheme of parabolic OCP,the lowest order Brezi-Douglas-Marini mixed finite element space（abbreviated as BDM₁ space）is used to approximate the state variables,and the piecework constant function is used to approximate the control variables.Strictly derive the ²L error of the state and control variables,as well as the first-order convergence of the space.When the fully discrete scheme of parabolic OCP is studied by MFMFE method,the state variables are spatially discretized by MFMFE and temporally discretized by backward Euler difference method.The state variables are approximated using BDM₁space,and the control variables are approximated using piecewise constant functions.Theoretical analysis obtains prior errors in state and control variables,as well as first-order convergence in time and space.Finally,a numerical example is given to verify the rationality of the results.