Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is fairly simple for particle methods, e.g., Nystrom-type discretizations, but more challenging if the supports of basis functions have to be taken into account. In this article, we present a modified construction for $\mathcal{H}^2$-matrices that uses translation-invariance to significantly reduce the storage requirements. Due to the uniformity of the boxes used for the construction, we need only a few uncomplicated assumptions to prove estimates for the resulting storage complexity.

翻译：椭圆偏微分方程的边界元方法通常会产生具有平移不变核函数的边界积分算子。对于粒子方法（例如Nyström型离散化），利用这一性质相当简单，但如果必须考虑基函数的支撑集，则更具挑战性。在本文中，我们提出了一种改进的$\mathcal{H}^2$矩阵构造方法，该方法利用平移不变性显著降低了存储需求。由于构造中使用的网格单元具有均匀性，我们仅需几个简单的假设即可证明所得存储复杂度的估计。