重椭圆方程的弱有限元方法研究

项目名称： 重椭圆方程的弱有限元方法研究

项目编号： No.11526113

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王春梅

作者单位： 南京师范大学

项目金额： 3万元

中文摘要： 弱有限元(WG)方法最早在2011年由Junping Wang和Xiu Ye提出, 是求解偏微分方程数值解的一种新型高效的数值方法. 重椭圆方程起源于荧光分子断层成像(FMT)模型, 形式上可看作广义重调和方程, 但和重调和方程存在本质的区别. FMT是使用活体非侵害深度分辨局部化和量化荧光标记夹杂物的一种新兴的三维光学成像模式. FMT广泛地应用在早期癌症探测, 肿瘤切除术以及药物监测和开发. 由于经典的有限元方法尚不能有效求解此重椭圆方程, 本项目拟研究求解重椭圆方程的WG方法, 杂交WG(HWG)方法, 及其快速算法和超收敛性, 以促进FMT在医学, 生物和工程等领域的应用. 本项目拟从以下方面开展研究: (1)研究求解重椭圆方程的WG方法; (2)研究所提出WG方法的稳定性, 收敛性以及超收敛性; (3)研究求解重椭圆方程的HWG方法; (4)研究所提出WG/HWG方法的快速算法.

中文关键词： 荧光分子断层成像；弱有限元方法；有限元方法；重椭圆方程；正交解和核校正方法

英文摘要： Weak Galerkin(WG) finite element method was first proposed by Junping Wang and Xiu Ye in 2011, which is a new and efficient numerical method for solving PDEs. The bi-elliptic equation arises from fluorescence molecular tomography(FMT) model, which can be seen as a generalized biharmonic equation. However, the bi-elliptic equation differs from the biharmonic equation essentially. FMT is an emerging three-dimensional optical imaging modality which uses invivo noninvasive depth-resolved localization and quantification of fluorescent-tagged inclusions. FMT techniques have been extensively employed in early cancer detection, guidance of tumor resection as well as drug monitoring and discovery. The significance of the proposed research includes development of finite element methods for the bi-elliptic equation for which no existing method works. The goal of this proposal is to innovate and analyze efficient, stable and fast numerical methods---weak Galerkin (WG) methods, hybridized WG(HWG) methods, as well as their fast algorithms and superconvergence to address challenges in the bi-elliptic equation so as to empower biologists and engineers conducting fundamental and applied research in this area. The intellectual merits of the proposed work consist of: (1) Developing robust weak Galerkin finite element methods for t

英文关键词： Fluorescence Molecular Tomography；weak Galerkin finite element method；finite element method；bi-elliptic equation；orthogonal solution and kernel correction algorith