Modern numerical methods for partial differential equations

偏微分方程是科学和工程中最重要的研究对象之一。这些方程的解将揭示正在研究的系统的性质。通常，这些方程表示质量、能量和动量守恒。例如，飞机制造商对设计最大限度地提高效率、性能和安全性的飞机机翼感兴趣。相关的偏微分方程组是上面提到的未知速度场和压力场的守恒定律。不幸的是，这些方程的显式解非常罕见。一般而言，科学家和工程师依靠数值方法来求解偏微分方程组。在真实的3D情况下，可能有数百万个未知数和方程式。最强大的计算机不能使用经典方法在合理的时间内求解这些方程。

Partial differential equations (PDEs) are among the most important objects of study in science and engineering. Solution of these equations would reveal the properties of the system under study. Very often, these equations express conservations of mass, energy and momenta. For instance, aircraft manufacturers are interested in designing wings of aircraft maximizing efficiency, performance and safety. The relevant PDEs are the conservation laws mentioned above for the unknown velocity and pressure fields. Unfortunately, explicit solutions of these equations are very rare. In general scientists and engineers rely on numerical methods to solve PDEs. In realistic 3D cases, there may be many millions of unknowns and equations. The most powerful computers are not able to solve these equations in reasonable time using classical methods.