Numerical Solution of Partial Differential Equations

Numerical solution of PDEs is rich and active field of modern applied mathematics. The steady growth of the subject is stimulated by everincreasing demands from the natural sciences, engineering and economics to provide accurate and reliable approximations to mathematical models involving partial differential equations (PDEs) whose exact solutions are either too complicated to determine in closed form or, in many cases, are not known to exist. While the history of numerical solution of ordinary differential equations is firmly rooted in 18th and 19th century mathematics, the mathematical foundations of the field of numerical solution of PDEs are much more recent: they were first formulated in the landmark paper Über die partiellen Differenzengleichungen der mathematischen Physik (On the partial difference equations of mathematical physics) by Richard Courant, Karl Friedrichs, and Hans Lewy, published in 1928. There is a vast array of powerful numerical techniques for specific PDEs: level set and fast-marching methods for front-tracking and interface problems; numerical methods for PDEs on, possibly evolving, manifolds; immersed boundary methods; mesh-free methods; particle methods; vortex methods; various numerical homogenization methods and specialized numerical techniques for multiscale problems; wavelet-based multiresolution methods; sparse finite difference/finite element methods, greedy algorithms and tensorial methods for high-dimensional PDEs; domaindecomposition methods for geometrically complex problems, and numerical methods for PDEs with stochastic coefficients that feature in a number of applications, including uncertainty quantification problems. Our brief review cannot do justice to this huge and rapidly evolving subject. We shall therefore confine ourselves to the most standard and well-established techniques for the numerical solution of PDEs: finite difference methods, finite element methods, finite volume methods and spectral methods. Before embarking on our survey, it is appropriate to take a brief excursion into the theory of PDEs in order to fix the relevant notational conventions and to describe some typical model problems.