High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral methods. These tools make it difficult to apply these techniques to higher dimensions. In contrast, fixed Cartesian grid methods, such as the immersed boundary (IB) method, are easy to apply and generalize, but typically are low-order accurate. In this study, we introduce the Smooth Forcing Extension (SFE) method, a fixed Cartesian grid technique that builds on the insights of the IB method, and allows one to obtain arbitrary orders of accuracy. Our approach relies on a novel Fourier continuation method to compute extensions of the inhomogeneous terms to any desired regularity. This is combined with the highly accurate Non-Uniform Fast Fourier Transform for interpolation operations to yield a fast and robust method. Numerical tests confirm that the technique performs precisely as expected on one-dimensional test problems. In higher dimensions, the performance is even better, in some cases yielding sub-geometric convergence. We also demonstrate how this technique can be applied to solving parabolic problems and for computing the eigenvalues of elliptic operators on general domains, in the process illustrating its stability and amenability to generalization.

用于求解任意域上的椭圆方程的高阶数值方法通常需要专门的设备，例如用于有限元方法的高质量合规网格和用于边界积分方法的正交规则。这些工具使得很难将这些技术应用于更高的维度。相反，固定的笛卡尔网格方法（例如，沉浸边界（IB）方法）易于应用和推广，但通常精度较低。在这项研究中，我们介绍了平滑强迫扩展（SFE）方法，这是一种固定的笛卡尔网格技术，该技术建立在IB方法的洞察力之上，可以使人们获得任意精度的顺序。我们的方法依靠一种新颖的傅立叶连续法来计算不均匀项到任何所需正则性的扩展。结合高精度的非均匀快速傅立叶变换进行插值运算，从而得出一种快速而可靠的方法。数值测试证实，该技术在一维测试问题上的表现与预期的一样。在更高的尺寸上，性能甚至更好，在某些情况下会产生次几何收敛。我们还演示了该技术如何在解决抛物线问题和计算椭圆形算子在一般域上的特征值的过程中，说明其稳定性和适用性。在某些情况下会产生次几何收敛。我们还演示了该技术如何在解决抛物线问题和计算椭圆形算子在一般域上的特征值的过程中，说明其稳定性和适用性。在某些情况下会产生次几何收敛。我们还演示了该技术如何在解决抛物线问题和计算椭圆形算子在一般域上的特征值的过程中，说明其稳定性和适用性。