=ADD=
=reftype=
14
=number=
03-10
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2003/03-10.ps.gz
=year=
2003
=month=
06
=author=
Gu; Hong
=title=
Graphical Solutions to the Plateau Problem
=abstract=
There already exist a variety of software for generating minimal
surfaces of special types. However, the convergence theories for those
approximating techniques are always left uncompleted. This leads to the
difficulty of  programming for displaying a whole class of minimal surfaces
in general form. 
In the old years, some research work has been done for solving the well-known
Plateau Problems by the finite element method,
under the assumption that 
the associated partial differential equation contains unique solution
for some special domain and boundary conditions.
In this thesis, we try to extend the existing convergence theories to
some new discretization form, which is convenient for symbolic
computation, and finally generate those 
approximated minimal surfaces (subject to the 
Plateau problems) by aid of computer algebra software.

The key boundary value differential form based on zero mean 
curvature will be given first and,
for solving the associated discrete scheme, either the
numerical Newton iteration method can be applied, 
or we can use some symbolic solving methods for the parameter
depending discrete equations. 
Compared to other existing finite element approaches for
solving the Plateau problems, 
the new special discretization methods in this paper have many convenient 
properties for applying symbolic approaches.  
The convergence and superconvergence property of the numerical solutions
are extended based on the new discretization forms on general
finite element interpolation spaces.
This method can be applied to generating the
minimal surface graphically on specialized software. And, in order to
speed up the whole computation, some 2-grid discretization algorithms
and parallel algorithms 
have already been proved and implemented into the program.    
=note=
PhD Thesis
=sponsor=
FWF project SFB013/1304
=keywords=
Minimal surface, Plateau problem, Finite element method, Sobolev space, Variational form, Convexity, Brouwers fixed point theorem, 2-grid discretization, Parallel algorithms