High Efficient Numerical Methods For Inverse Stefan Problems With Domain Embedding Methods

This thesis is intended for developing efficient numerical methods for inverse Stefan problems with domain embedding methods.The problem is reformulated as finding p ? H1(?2×(0,T))such that y(p)(x,t)satisfies y(p)(x,t)=g(s,t),(s,t)??1×(0,T),and is also the solution of the following problem:(?)where D is the fixed domain,?1(fixed)and ?2(fictitious)are the boundaries of D,respectively.Due to the ill-posedness of the above problem,it is reformulated as a minimization problem in terms of the Tikhonov regularization method.Based on the existing framework of finite element discretization for the above minimization problem,we present the detailed procedures for implementing the numerical method,with the emphasis on solving the minimization problem by the conjugate gradient method combined with computing the numerical solution of the related forward problem step by step in time,in order to increase the computational efficiency.We also use the two-parameter method based on the Morozov discrepancy principle to determine the regularization parameter,which is very useful in actual applications.Numerical experiments are provided to show the computational performance of the methods discussed in this thesis.