多分辨分析法解类Stefan问题

Solid-liquid phase transition is an important subject in the study of free boundary problem. This study proposes a new type of efficient numerical methods called fictitious domain/wavelets method. We mainly focus on how to precisely capture the moving boundary and solve the heat equation defined on each phase. Compared with conventional space discretization methods such as finite difference method and finite element method, the wavelets method has the following advantages: adaptive approximation，connection with multiresolution analysis, preconditioning to the large system arising from the discretization of partial differential equation, fast algorithms related to wavelets and characterization of function space. In this project, we do a comprehensive exploration of free boundary problem with wavelets method: 1. We discusse the existence and uniqueness of solution of elliptic problem on genenral domain which is obtained from the implicit time discretization of heat equation. Here we use Petrov-Galerkin approximation method with wavelets bases to have unit stiff matrix in order to facilitate the numerical implementation. Our main concern is the the compatibility of the discretizaiton space on the fictitious domain and on the boundary , i.e. the LBB condition. Usually the discretization level on the former is greater than that on the latter to some extent. 2. How to upgrade the regularity of solution around the original boundary. There are two approches. One is a smooth fictitious domain method which defines the Lagrange multipliers on the control boundary that has a positive distance with the original boundary in the outer normal direction. Another is smooth extension method based on the construction of multiresolution analysis on the interval. It involves the regularity of wavelets themsevles. 3. We establish the error analysis , mainly interior error. Note the different conditions under with we conclude the interior error estimate. They are Petrov-Galerkin method and non-compactly support bases. 4. We give condition number analysis and precondition the large system after wavelets discretization. We try to verify the fact that the irrelevance between condition number and discretization level on fictitious domain that has been observed in the numerical examples. 5. We demonstrate some numerical simulations of Stefan problem with the Dirichlet and Neumamm boundary conditions. The boundary condition is of type Gibbs-Thomson which contains the curvature and velocity of boundary.

固体-液体的相转变是研究移动边界问题中的一个重要课题。本研究提出一种新型的高效的数值计算方法-即虚拟区域/小波法。我们主要围绕如何精确求解移动边界的位置和各相上的热方程进行展开。与传统的差分法和有限元法进行空间离散相比，小波基的应用具有如下优点：自适应近似性、与多分辨分析方法的联系、方程离散后大型系统的预条件处理、相关的快速数值算法和对范函空间的刻划。在本项目中我们对用小波法处理移动边界问题进行了全面的探索：1.我们讨论热方程时间离散后解的存在唯一性。我们主要关心在虚拟域和边界上的离散空间的相容性，即LBB条件。2.如何提高方程解在边界附近的光滑性。我们建立了光滑虚拟法和以多分辨分析构造为基础的函数光滑延拓法。3.我们将建立解的误差分析,主要是内部误差。4.我们对离散产生的系统进行条件数分析和预条件处理。5.数值模拟具有Dirichlet和Neumamm边界条件的Stefan问题。

由于多分辨分析法在数值计算中的优点，如通过提高尺度来提高相应的计算精度，存在重要的快速算法和对方程离散后矩阵条件数的预处理，此方法在偏微分方程数值计算中占着越来越重要的地位。此项目针对在一般区域上构造小波基的局限性，我们提出一种新型的耦合多分辨分析和虚拟区域法解椭圆型方程的数值算法。为验证此方法的有效性，我们把它应用到移动边界中的一个重要课题Stefan问题。此项目的主要工作成果包括：1. 椭圆型方程的连续和离散解的存在和唯一性证明；2.为达到此方法的最优收敛速度，我们在函数的光滑延拓和内部误差的理论证明这两方面进行研究并在数值例子中得到验证；3.应用此方法到带Dirichlet移动边界的Stefan问题。我们通过此项目的研究，为应用多分辨分析法解移动边界问题奠定了基础。在此项目研究基础上，我们可进一步讨论多分辨分析法与传统离散法相比在可适应性和条件预处理中的优点，并推广到带更复杂边界条件的问题中。