Spectrally accurate solutions to inhomogeneous elliptic PDE in smooth geometries using function intension

Abstract

We present a spectrally accurate embedded boundary method for solving linear, inhomogeneous, elliptic partial differential equations (PDE) in general smooth geometries in two dimensions, focusing in this manuscript on the Poisson, modified Helmholtz, and Stokes equations. Unlike several recently proposed methods which rely on function extension, we propose a method which instead utilizes function intension, or the smooth truncation of known function values. Similar to those methods based on extension, once the inhomogeneity is truncated we may solve the PDE using any of the many simple, fast, and robust solvers that have been developed for regular grids on simple domains. Function intension is inherently stable, as are all steps in the proposed solution method, and can be used on domains which do not readily admit extensions. We pay a price in exchange for improved stability and flexibility: in addition to solving the PDE on the regular domain, we must additionally (1) solve the PDE on a small auxiliary domain that is fitted to the boundary, and (2) ensure consistency of the solution across the interface between this auxiliary domain and the rest of the physical domain. We show how these tasks may be accomplished efficiently (in both the asymptotic and practical sense), and compare convergence to two recent high-order embedded boundary schemes. Finally, to demonstrate the wide applicability of the method, we solve a nonlinear predator-prey model, achieving rapid convergence in both space and time.