A number of recent results, including special discretization schemes,
adaptive methods and multilevel iterative methods for the resulting
algebraic systems, will be presented in this talk for various partial
differential equations (PDE). With a careful and combined use of
qualitative properties of PDEs, the underlying functional spaces and their
discretizations, many different kinds of equations will be treated with
the same or similar techniques. After an introduction to some practically
efficient methods such as the algebraic multigrid method for the Poisson
equations, it will be shown how more complicated systems such as linear
elasticity equations, electro-magnetic equations, porous media, Stokes
equations and more general newtoninan-nonnewtonian models can be reduced
to the solution of a sequence of Poisson equation and its simple variants.
The efficiency of these algorithms will be illustrated by theoretical
analysis, numerical examples and engineering applications.