We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a simply connected, planar domain. The eigenvalues are the characteristic values of layer potential operators involving the Hankel function. First, we obtain an asymptotic formula for the Laplace eigenvalues with respect to the perturbation of the domain. The results are based on the Gohberg--Sigal theory for operator valued functions. Second, we propose a numerical computation scheme to compute the Laplacian eigenvalues based on the matrix formulation of the layer potential operators with a geometric basis associated with the exterior conformal mapping of the domain. This provides a way to compute Laplace eigenvalues, $\lambda$, by finding roots of polynomials in $\lambda$ and $\log\lambda$. We also derive a fully computable a priori error estimate with no assumption on the domain's convexity. This shows the relation between the domain's regularity and the convergence rate of the proposed method to compute Laplace eigenvalues.
翻译:我们考虑Laplacian 的 Dirichlet 和 Neumann 和 Neumann 等值, 用于简单连接的平面域。 egen 值是涉及 Hankel 函数的层潜在操作员的特性值。 首先, 我们获得关于域扰动的 Laplace 和 egen 值的无线配方公式。 结果基于操作员有价值功能的Gohberg- Sigal 理论 。 第二, 我们提出一个数字计算方案, 计算 Laplace 和 egen 值, 以层潜在操作员的矩阵公式为基础, 与域外部符合映射相联的几何基数基值。 这为计算 Laplace egen值( $\ lambda$) 提供了一种方法, 以 $\ lambda$ 和 $\ log\ lambda$ 来找到多元值的根基值。 我们还得出一个完全可比较的先验误估值, 而不假定域的共性。 。 这显示了域域域域域域的规律性和折合率值方法的趋合率 。 。 这显示了域的规律性和折合拉基值 。