The accurate and efficient evaluation of potentials is of great importance for the numerical solution of partial differential equations. When the integration domain of the potential is irregular and is discretized by an unstructured mesh, the function spaces of near field and self-interactions are non-compact, and, thus, their computations cannot be easily accelerated. In this paper, we propose three novel and complementary techniques for accelerating the evaluation of potentials over unstructured meshes. Firstly, we rigorously characterize the geometry of the near field, and show that this analysis can be used to eliminate all the unnecessary near field interaction computations. Secondly, as the near field can be made arbitrarily small by increasing the order of the far field quadrature rule, the expensive near field interaction computation can be efficiently offloaded onto the FMM-based far field interaction computation, which leverages the computational efficiency of highly optimized parallel FMM libraries. Finally, we show that a separate interpolation mesh that is staggered to the quadrature mesh dramatically reduces the cost of constructing the interpolants. Besides these contributions, we present a robust and extensible framework for the evaluation and interpolation of 2-D volume potentials over complicated geometries. We demonstrate the effectiveness of the techniques with several numerical experiments.
翻译:准确和高效地评估潜力对于部分差异方程式的数值解决方案非常重要。 当潜力的整合领域不规则,并且由无结构的网格分离时, 近场和自我互动的功能空间是非对称的, 因此, 计算不易加速。 在本文中, 我们提出三种新颖和互补的技巧, 以加速评估无结构的网目的潜力。 首先, 我们严格地描述近场的几何学, 并表明这一分析可以用来消除所有不必要的近场互动计算。 其次, 由于近场可以通过增加远场四边规则的顺序而任意缩小, 近场互动的功能空间可以有效地卸载到基于FMM的远场互动计算中, 从而利用高度优化的平行的FMMM图书馆的计算效率。 最后, 我们显示, 与二次网格网格模型错开的单独的内插图解图, 将大幅降低构建内部平台的成本。 其次, 我们展示了多个复杂数字实验框架 。