Error Estimates And Preconditionings For Singular Perturbation And Optimal Control Problems

With the deepening of scientific research and engineering technology,a large number of natural phenomena in nature and many economic and social phenom-ena in daily life can often be described with the help of(partial)differential equations.Due to the influence of many factors,it is difficult to get the real solution of science and engineering problems.Scientific computing is one of the important scientific and technological progress in the past two centuries.It has become an important means to promote major scientific discoveries and scientif-ic and technological progress.It is also a key element of national scientific and teclmological innovation and development.Scientific computing must rely on ef-ficient numerical calculation methods and high-performance computer hardware systems.However,the enhance speed of computer hardware technology can not keep up with the development of science and engineering to a certain extent,so we must rely on the research and design of officient numerical methods for nu-merical simulation of large-scale engineering problems,and this is also one of the most effective and cost-effective solutions.The core of the adaptive technique is to construct an effective indicator of a posteriori error estimation by using the existing numerical results and the known information of the model equation.How to get an effective indicator of a posteriori error estimation,which is convenient for program implementation,is one of the focuses of many researchers' discussion and research.In addition,the optimal control models of integer order and fractional order partial differential equations for the optimization of control system performance index can be summarized as the problem of finding the extremum of objective function under a set of equality or inequality constraints.Due to the global characteristics of fractional derivative operator,many scholars at home and abroad use spectral method to solve variable constrained fractional optimal control problem.In this paper,with the finite element method,the numerical solution method of the integer order optimal control problem with variable constraints and the re-lated problems of the fast calculation of the discrete algebra system are discussed.Combined with the structural characteristics of the equivalent discrete algebra-ic equations,efficient block diagonal preprocessors arc constructed.The discrete scheme of the state variable integral limited fractional order optimal control prob-lem is given by using the spectral method,and the discrete scheme of the state variable integral limited fractional order optimal control problem is realized.In addition,the spectral method is used to solve the singular perturbation problem.The techniques for posteriori error estimation of spectral methods are discussed according to the orthogonal characteristics of the basic function.It includes the following contents.Considering the model of singularly perturbed reaction-diffusion equation in one-dimensional domain,we design orthogonal basis functions including singular-ly perturbed parameters by using weighted orthogonal generalized Jacobi poly-nomials.The corresponding numerical solution scheme of one-dimensional singu-laxly perturbed problem model is given based on spectral method.Based on the differential operator of the model equation,the relationship between the coeffi-cients of numerical solution and the right end term of the equation is established.Based on the weighted orthogonality of basis functions and generalized Jacobian polynomials,by analyzing the upper bound estimates of orthogonal coefficients of basis functions,two kinds of posteriori error estimates in norm sense are given with details.Based on the integral constraints satisfied by the control variables,the equiv-alent optimality conditions of the distributed optimal control problem are given,and the numerical discrete algebraic system of the model problem is given by us-ing the finite element method.According to the structural characteristics of the non-zero elements in the stiffness matrix,a robust block preprocessor is designed,and a fast iterative algorithm is designed.At the same time,the computational complexity of the algorithm is analyzed.Finally,a numerical example is given to verify the efficiency of the preprocessor,and the computational complexity of the corresponding iterative algorithm is in line with the theoretical analysis results.Similarly,around the elliptic optimal control problem with state variables in the sense of integral constraints,the first-order equivalent optimality condition is given by using KKT condition,and the numerical discretization of the corre-sponding equivalent problem is realized by using finite element method.At the same time,according to the structural characteristics of its stiffness matrix,a robust block preprocessor and a feasible iterative algorithm are designed.Ac-cording to the similar case of integral constraint of control variable,it is proved that the iterative computation is 6 steps.Similarly,a numerical example is given to verify the efficiency of the preprocessor,and the computational complexity of the iterative algorithm is consistent with the theoretical analysis results.In addition,by introducing Lagrange multiplier technique,the first-order optimal-ity condition of optimal control problem with state variable constrained in the sense of L2-norm is analyzed,and the equality correspondence between control variable and dual state variable is obtained.In addition,for the fractional order partial differential equations in the sense of Riemann-Liouville,the corresponding optimality conditions of the Riesz frac-tional order optimal control problem model with state variables under integral constraints are investigated in detail.With the help of the global characteris-tics of Galcrkin spectral method and the generalized Jacobian polynomial,the Galerkin spectral method is constructed to discretize the fractional order optimal control problem model.At the same time,a priori error estimation analysis of the numerical solution of the model is given according to the existing regularity analysis results.Finally,the approximation effect of the high-precision Galerkin spectral method numerical scheme is verified by numerical examples,and the correctness of the theoretical results is further verified by the convergence order analysis of the numerical solution.