曲率多边形的Trefftz有限元
Trefftz Finite Elements on Curvilinear Polygons
摘要：
提出了一种曲多边形网格上的Trefftz型有限元法。使用积分方程技术计算局部基函数，该积分方程技术允许对形成局部刚度矩阵所需的量进行有效且准确的评估。为了在存在曲边的情况下定义我们的局部有限元空间，我们还必须适当地定义定义在曲边上定义的函数在该边上是给定次数的“多项式”意味着什么。我们考虑两个自然的选择，在解决的一个，产生完整的多项式空间包含在我们的本地有限元空间，并讨论如何在实践中与这些边缘多项式空间。对所得到的有限元引入了一个插值算子，并证明了在合理的假设下，该插值算子对插值误差具有最优阶收敛性。我们提供了一个描述的积分方程方法用于本文中的例子，这是最近开发的，正是考虑到这些应用。几个数值例子说明了这种最优阶收敛的有限元解的某些家庭的网格中，每个元素至少有一个曲边。我们还表明，它是可能的，以实现最佳的顺序收敛，而无需典型的自适应细化对奇点利用局部奇异函数的逼近能力，可能存在于我们的有限元空间。
英文摘要：
We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local finite element spaces, and discuss how to work with these edge polynomial spaces in practice. An interpolation operator is introduced for the resulting finite elements, and we prove that it provides optimal order convergence for interpolation error under reasonable assumptions. We provide a description of the integral equation approach used for the examples in this paper, which was recently developed precisely with these applications in mind. A few numerical examples illustrate this optimal order convergence of the finite element solution on some families of meshes in which every element has at least one curved edge. We also demonstrate that it is possible to exploit the approximation power of locally singular functions that may exist in our finite element spaces in order to achieve optimal order convergence without the typical adaptive refinement toward singular points.
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