A CutFEM Method For An Optimal Control Problem Governed By Wave Equations With Interface

Many mathematical models in the fields of science and engineering can be transformed into optimal interface control problems governed by partial dif-ferential equations.For example,in the propagation of ocean acoustic waves,seismic waves,and electromagnetic waves,due to the different properties of the materials on both sides of the inner interface of the calculation area,the coefficients of the wave equation's diffusion term will be discontinuous,and the waves will have discontinuities on both sides of the inner interface.Different values produce jumps in the normal direction of the inner boundary.From another perspective.the normal jump can be adjusted to affect the wave prop-agation and achieve the expected wave field distribution effect.This paper studies the CutFEM method for an optimal control problem governed by wave equation with interface.The main contents are as follows:First,the model for the optimal control problem governed by the wave equation with inner interface is established,and then the adjoint state equa-tion and optimality condition inequality of the model are derived in detail based on the classical optimal control problem theory.The original state equa-tion along with the adjoint state equation and the optimality inequality,these three formulas are combined to form the required optimality system.Then,the CutFEM method we adopt for the spatial discretization of the optimal system is introduced specificly.The CutFEM method is an unfitted finite cl-ement method based on Nietzsche's method proposed by Peter Hansbo and Anita Hansbo.The difference between the unfitted finite element method and other finite element methods is that in the unfitted finite element method,the internal interface can cross the edge of the cell,which can avoid complicated interface partition.For the discretization on the time level of the optimality system,we adopt the Crank-Nicolson time second-order discretization scheme.In this way,we finally get the fully discrete scheme of the optimality system.In order to derive the error estimates,the paper introduces the interme-diate variables of the state variable and the adjoint state variable,and then uses the intermediate variable as a bridge to derive the error estimate.By constructing error equations,using specific error analysis methods,first we derive the error estimates between the fully discrete finite element solutions of the state variable and adjoint state variable and their intermediate variables.Then the error estimates between the fully discrete finite element,solution and the exact solution of the control variable is derived.Secondly,by defining an elliptic operator,the error estimates between the exact solutions of the state variable and adjoint state variable and their intermediate variables is derived.Finally,combining the above error estimates,we obtaine the error estimates between the exact solutions of the state variable,adjoint state variable and control variable and their fully discrete finite element solutions.At the end of the article,a preliminary numerical example was performed to verify the theoretical results.