Everything is more simple than one thinks but at the same time more
complex than one can understand Johann Wolfgang von Goethe To reach the
point that is unknown to you, you must take the road that is unknown to
you St. John of the Cross This is a book on the numerical approximation
ofpartial differential equations (PDEs). Its scope is to provide a
thorough illustration of numerical methods (especially those stemming
from the variational formulation of PDEs), carry out their stability and
convergence analysis, derive error bounds, and discuss the algorithmic
aspects relative to their implementation. A sound balancing of
theoretical analysis, description of algorithms and discussion of
applications is our primary concern. Many kinds of problems are
addressed: linear and nonlinear, steady and time-dependent, having
either smooth or non-smooth solutions. Besides model equations, we
consider a number of (initial-) boundary value problems of interest in
several fields of applications. Part I is devoted to the description and
analysis of general numerical methods for the discretization of partial
differential equations. A comprehensive theory of Galerkin methods and
its variants (Petrov- Galerkin and generalized Galerkin), as wellas
ofcollocationmethods, is devel- oped for the spatial discretization.
This theory is then specified to two numer- ical subspace realizations
of remarkable interest: the finite element method (conforming,
non-conforming, mixed, hybrid) and the spectral method (Leg- endre and
Chebyshev expansion).
