四阶偏微分方程的杂交间断有限元方法及自适应算法

项目名称： 四阶偏微分方程的杂交间断有限元方法及自适应算法

项目编号： No.11301396

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 黄学海

作者单位： 温州大学

项目金额： 23万元

中文摘要： 四阶偏微分方程不仅在组合弹性结构、相分离模型、图像去噪、静电驱动微机电系统、流体力学等领域有着重要应用，而且在一般多尺度问题高效求解算法中起关键作用。因此研究和构造四阶偏微分方程的高效数值求解方法具有重要的理论意义和直接的应用价值。本项目主要研究四阶偏微分方程的杂交间断有限元方法及自适应算法，包括：基于Hermann-Miyoshi形式，构造四阶偏微分方程具有最优收敛阶的间断有限元方法，对数值解进行后处理获得新的超收敛数值解；提出四阶偏微分方程杂交有限元方法的统一框架，刻画杂交化后的整体自由度并由此建立各种可杂交化有限元方法的内在联系，给出统一的先验误差分析，后处理杂交元法数值解获得新的超收敛数值解；构造四阶偏微分方程超收敛间断有限元方法和杂交有限元方法的后验误差估计子，设计自适应算法并开展拟最优收敛性和最优复杂度的理论分析研究。可望为四阶偏微分方程数值解的研究带来国际前沿的系统的新进展。

中文关键词： 四阶偏微分方程；杂交间断有限元方法；误差分析；自适应算法；后处理

英文摘要： The fourth-order partial differential equations are widely used in the fields of elastic multi-structures, phase separation models, image denoising, electrostatic micro-electromechanical systems, fluid mechanics, etc. Moreover, fourth-order partial differential equations also play a vital role in designing efficient numerical methods for general multiscale problems. Therefore, it is both theoretically and practically important to investigate efficient numerical methods for fourth-order partial differential equations. This project is intended to develop hybridizable discontinuous Galerkin finite element methods and associated adaptive algorithms for fourth-order partial differential equations. Firstly, optimal discontinuous Galerkin finite element methods based on Hermann-Miyoshi formulation will be proposed for fourth-order partial differential equations, and a new superconvergent numerical solution will be obtained by postprocessing. Then, we present a unified framework of hybridizable finite element methods for fourth-order partial differential equations and characterize the hybridized degrees of freedom, based on which some internal relations of various hybridizable finite element methods will be established. We also provide a unified a priori error analysis for these hybridizable methods and postprocess the

英文关键词： Fourth-Order Partial Differential Equations；Hybridizable Discontinuous Galerkin Methods；Error Analysis；Adaptive Algorithms；Postprocessing