We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of $O(N)$ for an adaptive discretization with $N$ degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip ``thickens'' an existing boundary quadrature.

翻译：我们提出了一种新的框架，用于在具有光滑边界的域中快速求解非齐次椭圆型边值问题。基于自适应盒码或快速傅里叶变换的高阶求解器可以高效处理体积非齐次性，但需要在边界附近小心处理，以确保体积数据全局光滑。我们通过将域划分为两个区域来避免边界附近的函数延拓或切割单元求积：一个是远离边界、可使用截断自由空间盒码高效处理的体区域，另一个是宽度可变、贴合边界的带状区域，该区域采用谱配置法及相应的快速直接求解器处理。随后，每个区域的特解与拉普拉斯层势相结合，得到全局解。对于具有N个自由度的自适应离散化，所得求解器具有$O(N)$的最优计算复杂度。通过高效的二维（2D）实现，我们展示了在跨越广泛长度尺度上对体积数据、边界数据和几何特征的自适应分辨率，通常能达到10位数字的精度。所有边界校正的成本相对于体盒码的成本保持较小。在许多情况下，向三维（3D）的扩展预计是直接的，因为该带状区域“增厚”了现有的边界求积。