A High-order Finite Element Method For Inverse Source Problem Of Second Order Elliptic Equations

In this paper, we discuss a high-order finite element method of minimal problem introduced by [22] for the inverse problem of the second-order elliptic differential equation, and establish the error estimation theory for two kinds of quadratic finite element solutions which is based on the global measurement data.The numerical experiments show that when the disturbance quantity is negligi-ble, the order of error for quadratic finite element solutions is higher than that for linear finite element; when the disturbance quantity is not negligible and the selection of some parameters are suitable, the decrease rate of the error with re-spect to disturbance quantity for quadratic finite element solution is higher than that for linear finite element. The experiment results verify the error estimation.Furthermore, we present two high-order finite element algorithms for solving in-verse source problem based on local measurement data. The numerical results show that the algorithms are stable and robust when the extension function of the local measurement data is smooth enough on the sub-region boundary.