论文标题
关于Laplace方程的离散，多边形域上的Neumann边界条件
On the discretization of Laplace's equation with Neumann boundary conditions on polygonal domains
论文作者
论文摘要
在本文中，我们描述了一类算法，用于在具有Neumann边界条件的多边形域上解决Laplace方程。众所周知，在这种情况下，解决方案在角落附近具有奇异性，这对许多现有方法构成了挑战。如果边界数据在多边形的每个边缘上平滑，则在每个角落附近，相应边界积分方程的解决方案在某些（分析可用）的单数幂方面具有扩展。使用解决方案的已知行为，已经为Dirichlet问题的解决方案构建了通用离散化。但是，根据$μ\ in（-1/2,0）$的$ o（t^μ）$的领先订单行为是$ O（t^μ）$，具体取决于角落的角度（相比之下，$ O（C+T^μ）$，带有$μ> 1/2 $，用于dirichlet问题）;这在普遍离散化的设计中提出了重大挑战。我们的方法是基于对Dirichlet问题的离散化，以通过求解伴随线性系统来计算“弱意义”的解决方案。也就是说，它可用于准确计算具有平滑功能的内部产品，但不能插值。此外，我们提出了一个程序，通过解决该角附近的一系列局部子问题，以任意接近角落获得准确的解决方案。结果用几个数值示例说明了结果。
In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which poses a challenge for many existing methods. If the boundary data is smooth on each edge of the polygon, then in the vicinity of each corner the solution to the corresponding boundary integral equation has an expansion in terms of certain (analytically available) singular powers. Using the known behavior of the solution, universal discretizations have been constructed for the solution of the Dirichlet problem. However, the leading order behavior of solutions to the Neumann problem is $O(t^μ)$ for $μ\in (-1/2,0)$ depending on the angle at the corner (compared to $O(C+t^μ)$ with $μ>1/2$ for the Dirichlet problem); this presents a significant challenge in the design of universal discretizations. Our approach is based on using the discretization for the Dirichlet problem in order to compute a solution in the "weak sense" by solving an adjoint linear system; namely, it can be used to compute inner products with smooth functions accurately, but it cannot be interpolated. Furthermore we present a procedure to obtain accurate solutions arbitrarily close to the corner, by solving a sequence of small local subproblems in the vicinity of that corner. The results are illustrated with several numerical examples.
