High Efficient Numerical Methods For Free Boundary Problems

Free boundary problems, for which the interface boundary has to be found as part of the solution, play fundamental roles in science and technology. Due to the high nonlinearity of such problems, it is almost impossible to get the closed forms of the solutions. Hence,numerical simulation is an important way in studying free boundary problems. The objective of this thesis is intended to create and analyze high efficient numerical methods for solving an inverse Stefan problem and an electro-thermal model for superconducting nanowire singlephoton detectors, and then carry out numerical experiments to illustrate the computational performance of the methods proposed.First of all, we study stability analysis of a class of finite element methods for initial boundary value problems of parabolic equations. Based on the continuous space-time finite element method given in [1], we derive the discrete method for solving initial boundary value problems of parabolic equations with Dirichlet boundary conditions or mixed boundary conditions. Then, by means of an intrinsic reasoning and technical derivation, we obtain the H1(QT)-norm stability estimates for the finite element solution. Such estimates are very useful in the area of inverse parabolic problems. As a matter of fact, it is our basis to analyze the algorithm for solving the inverse Stefan problem, developed later on.Secondly, we propose two domain embedding methods for solving an inverse Stefan problem. Following some ideas in [2], we enlarged the moving boundary domain ?(t) to a larger but simple and fixed domain ?. Then we introduce a Robin boundary condition on the fictitious boundary and use a new function q as a control variable instead of the unknown moving boundary s(t), from which we reformulate the original inverse problem as a linear operator equation. Thinking of the ill-posedness of the linear operator equation, we further transform it as an optimal problem in view of the regularization method, and then discretize it by a continuous space-time finite element method, to devise a numerical method for the inverse Stefan problem. We prove the unique solvability of the continuous and finite element problems and prove the finite element solutions converge to the solution of the continuous optimization problem. It should be pointed out that though the domain embedding method was proposed in [2] for solving one-dimensional inverse Stefan problems, no theoretical results were given there. we also devise a special space-time finite element method for solving one-dimensional inverse Stefan problems, which is easy to implement. Through a series of numerical experiments, we show the effectiveness of the methods proposed.At last, this thesis studies numerical simulation of an electro-thermal model for superconducting nanowire single-photon detectors(SNSPD). To the best of our knowledge, such work was not developed systematically in the literature. The model is is described as a nonlinear free boundary problem involving the temperature and the current, which are coupled together by a nonlinear parabolic interface equation and a second order ordinary differential equation. In this model, the SNSPD is approximated as a one-dimensional structure, the thermal response is modeled by a one-dimensional nonlinear parabolic interface equation involving the current flowing through the nanowire, we present Crank-Nicholson finite difference method and implicit-explicit scheme for the discretization. The electrical response is modeled by a second order ordinary differential equation. We rewrite it as a system of firstorder ordinary differential equations and present the trapezoidal rule for the discretization.We also use the idea of the shooting method combined with the phase transition condition to the determination of the interface position. A series of numerical experiments are provided to demonstrate the effectiveness of the method proposed.