Biot模型基于有限元离散的多重网格算法研究

项目名称： Biot模型基于有限元离散的多重网格算法研究

项目编号： No.11426189

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈罗平

作者单位： 西南交通大学

项目金额： 3万元

中文摘要： Biot模型描述了含流体的弹性多孔介质受外力挤压时的稳固化过程。该模型在建筑，环境及生物力学等领域具有很重要的应用价值。 Biot模型作为一种鞍点问题，其离散格式和快速算法的构造和分析对其它鞍点问题也有一定的借鉴意义。本项目旨在研究Biot模型稳定的有限元离散格式及相应代数方程组的多重网格算法。通过利用有限元外微分理论框架，我们构造Biot模型稳定的有限元离散格式并分析其收敛性。同时，通过分析相应离散系统的系数矩阵结构，设计出与网格尺寸无关的一致收敛的多重网格算法。我们还将研究将多重网格算法单独作为求解器及作为Krylov子空间迭代法的预处理子的数值表现，并分析算法的收敛性。

中文关键词： Biot固结方程；有限元方法；收敛阶；多重网格算法；一致收敛

英文摘要： Biot model describes the consolidation processes of a porous elastic medium which is partially saturated by a certain fluid when pressed by some external forces. Biot model has a wide range of applications such as in architecture, environment and biomechanics. As a saddle point problem, the development and analysis of its discretization schemes and fast solver methods can also be useful for other saddle point problems. This project aims to develop stable finite element discretizations for Biot model and multigrid methods for the corresponding algebraic system of equations. By applying the finite element exterior calculus framework, we will develop stable finite element discretizations for Biot model and analyze their convergence property. Moreover, by studying the matrix structures, we will construct uniform convergent multigrid algorithm that independent with the mesh sizes for the algebraic system. Furthermore, we will also study the performance of the multigrid algorithms as stand-alone solvers and as preconditioners for Krylov subspace methods both numerically and theoretically.

英文关键词： Biot consolidation equations；Finite element methods；Convergence order；multigrid methods；uniform convergence