Research On Linear Conforming Triangular Finite Element Method For Three Kinds Of Interface Problems

Interface problems arise from some physical situations when the research object contains two or more different media.In this case,the analytical solutions are difficult to be solved and the regularity is much lower,so it is very meaningful to study the corresponding numerical algorithm.In this dissertation,the linear triangular finite element method is employed for parabolic interface problems,hyperbolic interface problems,and elliptic interface control problems.On one hand,moving grids method is adopted to the above two types of time-dependent interface partial differential equations,Firstly,the finite element discretization is used to the space variables,and then the difference discretization is used to the time variables,but the finite element meshes can be different at different times.The optimal error estimate results are obtained without the traditional analysis tool of Riesz projection,which makes the proof process to be more concise.Some numerical results are given to verify the effectiveness of the proposed method.On the other hand,for the interface optimal control problem,based on the optimal conditional theory proposed by the famous mathematician Hinze,the linear triangular finite element method is proposed.The optimal order error estimates are obtained for the control,state and adjoint state.Since the linear finite element is the lowestorder finite element,it just meets the engineers preference,which is using the simples elements to obtain fine results.Therefore,the methods proposed in this dissertation have significant theoretical and application values for more complex interface problems in the following studies.