IGA And Proper Orthogonal Decomposition For Parameter Dependent Elliptic Problems

Isogeometric analysis(IGA)is a new numerical method based on spline theory,which is proposed based on the finite element method and aimed at achieving the integration between CAD and CAE.The spline functions used in the exact geometric representation of the computational domain are used as the basis functions of the trial function space.The computational domain is presenting geometrically exact and the mesh generation process is not needed in IGA.in addition,due to the high-order continuity of the basis functions used in IGA,it is more suitable in approximating the smooth solutions.Proper orthogonal decomposition(POD)based on Galerkin projection is a powerful method for generating low-dimensional models of dynamic systems with very large or even infinite dimensional phase space.In this paper we mainly investigate the combination of IGA and proper orthogonal decomposition based on Galerkin method for the model order reduction of linear and semi-linear elliptic equations.The research significance and status of IGA and POD are introduced firstly,and then give a review about B-spline and NURBS spline,Newton's method together with some definitions and theorems will be used in our discussion.Then we study the ideas and processes of IGA method around the second-order semi-linear elliptic equation with Neumann boundary condition,including the derivation of the corresponding variational problem,the existence and uniqueness of solutions to Variational problems,isogeometric spatial discretization and the error introduced by spatial discretization.Combined with the Newton iteration method,we study the calculation of matrix elements involved in solving the nonlinear equations obtained by Galerkin method,and complete a numerical program for solving nonlinear elliptic equation by IGA.The discrete POD method is discussed,and a numerical algorithm for calculating the POD basis function is proposed based on the representation of numerical solutions obtained by IGA method.The obtained POD basis functions are used as a basis to construct reduced-order model,then a smaller non-linear equations is obtained.The error estimation between the solutions of the reduced-order model and those of full-order models are given.The numerical tests of linear and semi-linear elliptic problems are provided to illustrate the effectiveness of IGA-based POD techniques in order reduction,and to verify the correctness of the error estimation proposed.