A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized representations of the boundary. For the Helmholtz equation, while many authors use numerical quadrature to evaluate these boundary integrals, we present analytical expressions for such integrals over flat polygons in the form of infinite series. These can be efficiently truncated based on the accurate error bounds, which is key to their integration in methods such as the Fast Multipole Method.
翻译:通过整体等式方法解决部分差分方程的一个关键问题是,对涉及Green的功能及其衍生物的单一组合体进行评估,将简单的函数乘以边界分立的表示法。对于赫尔姆霍尔茨等式,许多作者使用数字方形来评价这些边界组合体,我们用无限序列的形式对平面多边形的这种组合体提出分析表达方式。根据准确的误差界限,这些组合体可以有效缩短,而误差界限是将其纳入快速多极方法等方法的关键。