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The generalized finite element method: Numerical treatment of singularities, interfaces, and boundary conditions.

机译：广义有限元方法：奇异性，界面和边界条件的数值处理。

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摘要

This dissertation is devoted to numerical approximation of partial differential equations by Generalized Finite Element Method (GFEM), which is closely related to some other methods, such as the hp cloud method and the extended finite element method. As an extension of the standard Finite Element Method (FEM), the GFEM has more flexibility in dealing with complicated domain geometry, corner singularities, transmission problems and mixed boundary conditions. It differs from the standard FEM in the construction of the finite-dimensional space in which the approximate solution is sought. Instead of using piecewise polynomials on each element of a triangulation of the domain, the global GFEM space is defined by using partition of unity in combination with local approximation spaces defined in each patch of the partition. In this sense, the GFEM is an example of so called meshless methods, since the partition of unity need not be subordinated to a particular triangulation of the domain as the standard FEM does. The GFEM allows one to incorporate a priori knowledge of the local behavior of the solution in the construction of the approximation space, and gives the option of constructing trial spaces of any desired regularity.;For transmission (interface) problems on domains with smooth, curved boundaries, we establish quasi-optimal rate of convergence of the numerical solution to the true solution by using a non-conforming GFEM. To achieve this goal, we construct a sequence of approximation spaces Sn, satisfying the following two conditions: (1) nearly zero boundary and interface matching, (2) approximability. We then seek the numerical solution in this spaces as the Galerkin approximation to the true solution, and show that the approximation error of order O( hmn ), where hn is the typical size of elements in the GFEM space Sn, and m is the degree of polynomials used for the local approximation of the solution. Numerical experiments are presented to demonstrate these theoretical results.;We also study the GFEM approximation for Poisson problem in polygonal domains with corner singularities. It is well-known that the loss of regularity of the exact solution due to domain singularities will deteriorate the convergence rate of the standard FEM on quasi-uniform mesh. To circumvent this difficult, we pose the problem in certain weighted Sobolev spaces, and show that the continuous problem has the expected regularity in these spaces. We then construct GFEM approximation spaces using partition of unity and local approximation spaces. For the former, we use dilation techniques to deal with corner singularities, while we use standard piecewise polynomial spaces for the latter. We then establish quasioptimal rate of convergence of the GFEM approximation to the exact solution both in weighted Sobolev spaces and then in Hilbert spaces in terms of O(dim(Sn) --m/2), where dim(Sn) is the dimension of the GFEM space Sn.
机译：本论文致力于通过广义有限元法（GFEM）对偏微分方程进行数值逼近，该方法与hp cloud法和扩展有限元法等其他方法密切相关。作为标准有限元方法（FEM）的扩展，GFEM在处理复杂的域几何，拐角奇点，传输问题和混合边界条件时具有更大的灵活性。它与标准FEM的区别在于寻求近似解的有限维空间的构造。代替在域的三角剖分的每个元素上使用分段多项式，全局全局GFEM空间是通过使用单位分区与在分区的每个面片中定义的局部近似空间结合来定义的。从这个意义上讲，GFEM是所谓的无网格方法的一个示例，因为统一的划分不必像标准FEM那样服从域的特定三角剖分。 GFEM允许人们将近似解的先验知识整合到近似空间的构造中，并提供构建任何所需规则性的试验空间的选项。用于光滑，弯曲区域上的传输（界面）问题边界，我们通过使用不合格GFEM建立数值解与真解的准最优收敛速率。为了实现这个目标，我们构造了一系列满足以下两个条件的近似空间Sn：（1）边界和界面匹配几乎为零，（2）近似性。然后，我们在此空间中寻找数值解作为真实解的Galerkin近似值，并证明阶次为O（hmn）的近似误差，其中hn是GFEM空间中元素的典型尺寸Sn，m是度用于解决方案的局部逼近的多项式。通过数值实验证明了这些理论结果。我们还研究了具有角奇点的多边形区域中泊松问题的GFEM近似。众所周知，由于域奇异性而导致的精确解规则性的丧失将使标准FEM在准均匀网格上的收敛速度恶化。为了解决这一难题，我们在某些加权Sobolev空间中提出了问题，并证明了连续问题在这些空间中具有预期的规律性。然后，我们使用单位和局部逼近空间的划分来构造GFEM逼近空间。对于前者，我们使用膨胀技术来处理角奇点，而对于后者，我们使用标准的分段多项式空间。然后我们以O（dim（Sn）--m / 2）的形式在加权Sobolev空间中然后在希尔伯特空间中建立GFEM逼近到精确解的准最优收敛速度，其中dim（Sn）是GFEM空间Sn。

著录项

  • 作者

    Qu, Qingqin.;

  • 作者单位

    The Pennsylvania State University.;

  • 授予单位 The Pennsylvania State University.;
  • 学科 Applied mathematics.
  • 学位 Ph.D.
  • 年度 2012
  • 页码 94 p.
  • 总页数 94
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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