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.
翻译:解决任意领域等离子方程式的高顺序数字方法通常需要专门的机械,如对有限元素方法的高质量符合格格,和对边界整体法的二次规则。这些工具使得难以将这些技术应用于更高的尺寸。相反,固定的卡尔提斯格方法,如浸入边界法,容易应用和概括,但一般是低顺序精确的。在本研究中,我们采用平滑的叉路扩展法(SFE)方法,即固定的喀尔提斯格技术,该技术以IB方法的洞察为基础,允许人们获得任意的准确性定序。我们的方法依靠新型的Fourier继续方法来将不相容条件的扩展计算为任何预期的规律性。这与高度精确的不统一快速和稳健健健的方法相结合。我们用数字测试证实,该技术在一维测试问题上的精确性表现与预期的精确性。在更高层面,业绩甚至更好,在某些情况中产生次几何测地趋近的趋同性趋同性。我们还演示了该技术如何在一般的轨道上解决一般的稳定性。