We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in [Kublik, Tanushev, and Tsai - J. Comp. Phys. 247: 279-311, 2013]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.

翻译：我们利用[Kublik、Tanushev和Tsai-J.Comp.Phys. 247:279-311,2013]采用[Kublik、Tanushev和Tsai-J.Comp.Phys.247:279-311,2013]采用的方法,为边界整体操作者大家庭提出了新的更高阶梯。在这一配方中,一个光滑、封闭的超表层组成的边界被转化成一个相当的体积,在地表一个足够薄的管状周围定义。这种体积配方使得有可能在统一的笛轮网上使用简单的捕捉摸性规则,从而不必使用参数化来开发二次方形。因此,一个层的典型点奇特性在地表的正常线上延伸。我们建议对奇特点周围的网格上的捕捉住规则进行新的更高阶级更正。这一校正以奇特性的局部分解为基础,并取决于相对于地表的主要曲道方向的奇特性的角度。提议的分法与体形配,导致特殊的二次错误的取消。