重椭圆方程的弱有限元方法研究

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 the bi-elliptic equation; (2) Establishing a stability and convergence, including superconvergence, theory for the newly developed WG finite element methods by using new mathematical tools; (3) Proposing and analyzing a hybridized weak Galerkin finite element method for the bi-elliptic equation; (4) Developing the fast algorithms for the proposed WG/HWG methods.

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.

本项目研究了高效求解重椭圆方程的弱有限元(WG)方法。重椭圆方程起源于荧光分子断层成像(Fluorescence Molecular Tomography), 简称FMT, 模型. 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. Shui-Nee Chow, Ke Yin和Haomin Zhou于2013年提出了求解FMT模型的正交解和核校正方法(OSKCA), 在OSKCA中最关键的部分即为求解FMT模型的正交解, 而此问题即为求解重椭圆方程. 本项目采用的是近几年兴起的求解偏微分方程数值解的一种新型, 高效, 稳定的数值算法－弱有限元(WG)方法来数值求解此类重椭圆方程. 在WG方法的设计中, 可以使用任意网格(包括二维形状正则的任意多角形和三维形状正则的任意多面体)上的广义或间断近似函数来克服构造''光滑''有限元函数的困难. 在WG方法中, 通常可以通过引入其杂交形式或Schur补形式, 消去内部自由度, 生成一个仅仅依赖于边界自由度的规模缩小的线性方程组系统, 从而实现更高效, 更稳定的数值求解. 可以说, WG方法对偏微分方程的数值解法提出了一个全新的思想, 将经典的有限元方法提升到了一个更高的水平. 本项目提出了高效求解重椭圆方程的弱有限元方法，并给出了理论分析，并进行了相关的数值模拟，数值模拟结果和理论结果完全吻合，从而有效地求解了此类重椭圆方程。本项目的研究不仅能一定程度上填补偏微分方程数值解研究领域中求解此类重椭圆方程的空白, 而且能在一定程度上促进FMT在医学, 生物, 工程等各个领域的应用和发展, 从而有效地探测早期癌症以及监测和开发药物, 推动未来理解癌症和一大批其他疾病的分子原因和生物力学的发展.