论文标题
有关笛卡尔和分层网格的椭圆形部分微分方程的解决方案方法的概述
Overview of solution methods for elliptic partial differential equations on cartesian and hierarchical grids
论文作者
论文摘要
椭圆形的部分微分方程（PDE）在计算科学的许多领域都出现，例如计算流体动力学，生物物理学，工程，地球物理学等。由于其全球性质，有时是条件不足的操作员，它们很难解决。我们回顾了椭圆PDE的常见离散方法，例如有限差，有限体积，有限元和光谱方法以及它们形成的线性系统。我们还为这些离散化方法形成的线性系统提供了经典至现代解决方案方法的概述。这些方法包括分裂和Krylov方法，直接方法和分层方法。最后，我们展示的应用程序可以从椭圆PDE的快速有效的求解器中受益，包括不可压缩的Navier-Stokes方程的投影方法和带有分散校正的浅水波方程。
Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and sometimes ill-conditioned operators. We review common discretization methods for elliptic PDEs such as the finite difference, finite volume, finite element, and spectral methods and the linear systems they form. We also provide an overview of classic to modern solution methods for the linear systems formed by these discretization methods. These methods include splitting and Krylov methods, direct methods, and hierarchical methods. Finally, we show applications that would benefit from fast and efficient solvers for elliptic PDEs, including projection methods for the incompressible Navier-Stokes equations and the shallow water wave equations with dispersive corrections.
