基于谱正确有限元离散的弹性传输特征值问题高效数值方法

The transmission eigenvalues can be used to obtain estimates for the physical characteristics of the hidden scatterer and play a key role in the uniqueness and reconstruction in inverse scattering theory. The non-self-adjointness and nonlinearity plaguing the numerical study of the elastic transmission eigenvalue problem are compounded by the tensorial structure of the elastic wave equation. There exist only a few numerical studies and the theory is far from complete. Based on the fourth order formulation, this project aims to develop the spectrally correct finite element discretizations. Furthermore, the multi-level correction method is constructed to improve the efficiency. The study contents of this project include: (1) An iterative method based on the multi-level Bogner-Fox-Schmit element discretization is constructed and the accuracy of spectrum is analyzed. (2) Firstly, an equivalent and stable mixed scheme which admits the natural nested discretization is constructed. Then, we construct the multi-level correction method for the nested discretization on a series of nested grids to solve efficiently. And we show that this method is spectrally correct and prove the optimal convergence rate. (3) A multi-level correction algorithm based on the discontinuous Galerkin method is constructed. And the accuracy of spectrum and the optimal convergence rate are proved. The outcome of this project will improve the understanding of transmission eigenvalue problem of elastic waves, of which, the theory is still not sound. Besides, it will provide strong support for numerical research in various fields, such as medical diagnosis, geophysical exploration and seismic imaging.

传输特征值可用来估计未知散射体材料的物理性质，且对逆散射理论中唯一性及重构方法构造的发展有很重要作用。传输特征值问题的非自伴性、非线性性以及弹性波方程的张量结构，给弹性传输特征值问题的数值研究带来很大挑战。关于该问题理论分析和数值计算的工作都很少。本项目针对此问题的四阶形式，拟采用谱正确的有限元离散，进一步构造多重校正方法提高求解效率。研究内容包括：（1）构造基于多重Bogner-Fox-Schmit元离散的迭代方法，并分析其谱正确性；（2）构造一种跟原始问题完全等价、稳定、嵌套的混合格式，进一步构造多重校正方法提高求解效率，理论上分析该方法的谱正确性及最优收敛性等；（3）构造基于间断Galerkin方法的多重校正算法，并证明其谱正确性及最优收敛性等。本项目的研究将促进仍不健全的弹性传输特征值问题相关理论和应用的发展，将为医疗诊断，地球物理勘探，地震成像等领域数值研究提供有力支撑。

该项目主要研究了四阶特征值问题，尤其传输特征值问题的高效数值方法。传输特征值可用来估计未知散射体材料的物理性质，且对逆散射理论中唯一性及重构方法构造的发展有很重要作用。传输特征值问题的非自伴性、非线性性以及弹性波方程的张量结构，给弹性传输特征值问题的数值研究带来很大挑战。针对声波传输特征值问题，给出多连通区域上的多重混合元方法以及一种新的三次非协调元方法。针对弹性传输特征值问题，给出一种最低阶混合元方法、新的三次非协调元方法以及基于间断元离散的全纯Fredholm算子函数方法等。. 该项目主要创新点有：.1、对于四阶方程，首次给出新的三次非协调Bh03元基函数的显示表达式；.2、基于间断有限元离散，首次给出了一种新的全纯Fredholm算子函数方法。对于Laplace特征值问题，Biharmonic特征值问题以及弹性传输特征值问题，有完善的理论分析及数值模拟工作；.3、对于四阶特征值问题，在多连通区域上，首次给出了一种完全等价、稳定、嵌套的混合格式，进一步构造多重校正方法提高求解效率。.以第一作者身份发表高水平论文7篇；协助培养硕士研究生4人。