TABLE OF CONTENTS As is known, linear systems such as (1.1) often arise from solving other equations. Here we briefly review a selection of solution techniques for partial differential equations (PDEs) and nonlinear systems.
Many PDEs of practical interest are of second order. Solution techniques typically (though not always) reduce one order of differentiation before discretization via Gauss or Green s theorems or integration by parts. Below we discuss
the finite element method (FEM)
the boundary element method (BEM)
the finite difference method (FDM)
the finite volume method (FVM)
the global element methods (GEMs).
As each topic involves a huge subject, the reader will be referred to detailed references (and therein) for special perusal. We shall consider the interior Helmholtz-like equation as our model PDE (with p = ( x, y) ? ? ? 
2 and ?? smooth)
| (1.53) |  |
which is assumed to have a unique solution (for a suitable k).
The FEM is the most widely used method for solving PDEs since it was invented in 1950s and matured in 1970s; see [308,282,499]. For (1.53), it does not look for a so-called classical solution u ? C 2( ?) satisfying (1.53) partly because computer solutions are hardly in C 2( ?)anyway. Instead it seeks a weak solution u = u( x, y) ? V ? H 1( ?) such that
| (1.54) |  |
where
| (1.55) |  |
| (1.56) |
