界面问题和Laplace-Beltrami问题中的有限元超收敛及网格生成优化研究
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摘要
本文主要围绕二维椭圆界面问题和Laplace-Beltrami问题中的线性有限元的超收敛现象展开研究.对于二维椭圆界面问题,我们用界面拟合网格来离散求解区域,然后用标准线性有限元来离散其模型方程.在适度结构网格的假设下,考虑了界面拟合对超收敛的影响,证明了线性有限元解超逼近于真解的线性插值,并由此导出了二维椭圆界面问题的一个最大模的估计.我们还结合最新顶点加密算法和Borgers算法设计了一个二维界面拟合网格的快速生成算法Borgers算法是一个经典的二维界面拟合网格生成算法,其基本思想是在一致网格的基础上,把界面附近的网格结点扰动到界面上,然后对于界面附近被扰动的小四边形,选择适当的对角线来拟合界面和保持网格质量.对于适当光滑界面,可以证明Borgers算法生成的界面拟合网格是拟一致的和正则的.但对于结构复杂的界面,Borgers算法需要很密的一致网格来很好的拟合界面,这就使后续的计算浪费了很多不必要的计算量.改进后的Borgers算法,可以用较少的网格点更好地拟合复杂的界面,且能保证生成的网格的正则性.改进后的算法生成的界面拟合网格还有一定的结构性,可以利用现有的粗化算法得到一组嵌套网格.基于这组嵌套网格,我们设计了一个高效的多重网格解法器来求解相应的代数系统,数值试验表明这个解法器关于问题的规模和扩散系数的间断都是稳健的.我们的网格生成算法所用的数据结构,局部加密和粗化算法都非常简单,不需要继承树结构来存储网格数据.
对于Laplace-Beltrami (LB)司题,我们用三角形网格来逼近曲面,用曲面线性有限元来离散模型方程.同样在适度结构网格的假设下,考虑了用三角形网格逼近曲面带来的几何误差对超收敛的影响,证明了曲面线性有限元解超逼近于真解的线性插值.并把平面有限元中的几种梯度恢复格式推广到了曲面有限元上,其中包括简单平均,面积加权平均,全局和局部L2投影,以及Zienkiewicz和Zhu(ZZ)格式,并证明了恢复后的梯度可以更好的逼近真解的梯度.我们还考虑了曲面三角形网格的CVT优化问题,设计了一个足够光滑且能够反映曲面曲率变化的密度函数.实验表明这个密度函数用在曲面三角形网格的Lloyd优化方法中,可以有效改善网格的质量.
In this study, we focus on the superconvergence phenomenon in2D elliptic interface problem and Laplace-Beltrami problem.
For2D elliptic interface problem, adaptive mesh refinement and the Borgers'algo-rithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolant of the exact solution. Based on this superconvergence result, a maximal norm error estimate is obtained. The basic idea of the Borgers'algorithm is to use a Cartesian grid but perturb only the grid points near the interface onto the interface and choose an appropriate diagonal of perturbed quadrilaterals to fit the interface and maintain the mesh quality. The final body-fitted mesh is shape regular and topologically equivalent to the Cartesian grid. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.
For Laplace-Beltrami problem, superconvergence results and several gradient re-covery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the Laplace-Beltrami equation on general surfaces with mildly structured triangular meshes. For a large class of practically useful grids, the sur-face linear finite element solution is proven to be superclose to an interpolant of the exact solution of the Laplace-Beltrami equation, and as a result various postprocessing gradient recovery, including simple and weighted averaging, local and global L2-projections, and Zienkiewicz and Zhu (ZZ) schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experi-ments are presented to confirm the theoretical results. We also design a smooth density function which includes the surface curvature information and can be used in the CVT optimization of surface triangulation. The numerical tests show that the density function is effective to improve the quality of the surface triangulation.
对于Laplace-Beltrami (LB)司题,我们用三角形网格来逼近曲面,用曲面线性有限元来离散模型方程.同样在适度结构网格的假设下,考虑了用三角形网格逼近曲面带来的几何误差对超收敛的影响,证明了曲面线性有限元解超逼近于真解的线性插值.并把平面有限元中的几种梯度恢复格式推广到了曲面有限元上,其中包括简单平均,面积加权平均,全局和局部L2投影,以及Zienkiewicz和Zhu(ZZ)格式,并证明了恢复后的梯度可以更好的逼近真解的梯度.我们还考虑了曲面三角形网格的CVT优化问题,设计了一个足够光滑且能够反映曲面曲率变化的密度函数.实验表明这个密度函数用在曲面三角形网格的Lloyd优化方法中,可以有效改善网格的质量.
In this study, we focus on the superconvergence phenomenon in2D elliptic interface problem and Laplace-Beltrami problem.
For2D elliptic interface problem, adaptive mesh refinement and the Borgers'algo-rithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolant of the exact solution. Based on this superconvergence result, a maximal norm error estimate is obtained. The basic idea of the Borgers'algorithm is to use a Cartesian grid but perturb only the grid points near the interface onto the interface and choose an appropriate diagonal of perturbed quadrilaterals to fit the interface and maintain the mesh quality. The final body-fitted mesh is shape regular and topologically equivalent to the Cartesian grid. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.
For Laplace-Beltrami problem, superconvergence results and several gradient re-covery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the Laplace-Beltrami equation on general surfaces with mildly structured triangular meshes. For a large class of practically useful grids, the sur-face linear finite element solution is proven to be superclose to an interpolant of the exact solution of the Laplace-Beltrami equation, and as a result various postprocessing gradient recovery, including simple and weighted averaging, local and global L2-projections, and Zienkiewicz and Zhu (ZZ) schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experi-ments are presented to confirm the theoretical results. We also design a smooth density function which includes the surface curvature information and can be used in the CVT optimization of surface triangulation. The numerical tests show that the density function is effective to improve the quality of the surface triangulation.
引文
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