边界元方法的边界值修正
Boundary Value Correction for the Boundary Element Method
- University College London（伦敦大学学院）
- Jönköping University（延雪平大学）
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中文总结 AI 辅助
针对三维Dirichlet问题，提出一种边界元方法，通过沿法向的一阶Taylor展开修正替代边界上的边界值，利用Calderón通量实现无额外未知场的修正，在适当条件下达到\\(h^{k+1/2}\\)阶收敛，并通过球体和椭球体实验验证了其有效性。
中文摘要 AI 辅助
我们针对有界三维区域中带Dirichlet边界条件的Laplace方程，发展了一种边界元方法。该方法在内部替代边界上使用最大直径为\\(h\\)的平面单元，并沿单元法向通过一阶Taylor展开来修正施加的边界值。该修正利用了Calderón公式中已有的法向通量，不引入额外的未知场。在替代几何和边界算子的一致假设下，当\\(\gamma\delta_h\le c_0<4\\)时，惩罚形式是强制的，其中\\(\gamma\\)是惩罚参数，\\(\delta_h\\)是最大法向偏移。误差估计将迹逼近与大小为\\(\gamma^{1/2}\delta_h^2\\|u\\|_{H^3(\Omega)}\\)的几何项分开。当\\(\gamma\simeq h^{-1}\\)时，充分条件\\(\delta_h\lesssim h^{(k+1)/2}\\)在所述正则性和一致迹界下，对于\\(P_k/P_{k-1}\\) Dirichlet/通量空间产生\\(h^{k+1/2}\\)阶的误差。因此，\\(h^2\\)阶的偏移通过三次Dirichlet逼近支持该估计。我们分析了两种惩罚变体，并在球体和椭球体上进行了数值实验，以检验收敛性以及对几何、惩罚缩放和求积的敏感性。
英文摘要
We develop a boundary element method for the Laplace equation with Dirichlet boundary conditions in a bounded three-dimensional domain. The method uses flat panels of maximum diameter \(h\) on an interior surrogate boundary and corrects the imposed boundary values by a first-order Taylor expansion along the panel normals. The correction uses the normal flux already present in the Calderón formulation and introduces no additional unknown field. Under uniform assumptions on the surrogate geometry and boundary operators, the penalised form is coercive when \(γδ_h\le c_0<4\), where \(γ\) is the penalty parameter and \(δ_h\) is the maximum normal offset. The error estimate separates trace approximation from a geometry term of size \(γ^{1/2}δ_h^2\|u\|_{H^3(Ω)}\). With \(γ\simeq h^{-1}\), the sufficient condition \(δ_h\lesssim h^{(k+1)/2}\) yields an error of order \(h^{k+1/2}\) for \(P_k/P_{k-1}\) Dirichlet/flux spaces under the stated regularity and uniform trace bounds. Thus an offset of order \(h^2\) supports the estimate through cubic Dirichlet approximation. We analyse two penalty variants and present numerical experiments on a sphere and an ellipsoid to examine convergence and sensitivity to geometry, penalty scaling and quadrature.
