@**techreport**{RISC2464,author = {Hong Gu and Martin Burger},

title = {{Preprocessing for Finite Element Discretizations of Geometric Problems}},

language = {english},

abstract = {In this paper, we use finite element method to approximate the
solutions of parameter-dependent geometric problems, and
investigate the possibility of using symbolic methods as a preprocessing step.
The main idea of our approach is to construct suitable finite element
discretizations of the nonlinear elliptic equations leading to
systems of algebraic equations, which can be subsequently solved by
symbolic computation within the tolerance of computer algebra software.
The prolongation of the preprocessed symbolic
solution can serve as a starting value for a numerical iterative method on
a finer grid.
A motivation for this approach is that usual numerical iterations (e.g. via
Newton-type or fixed-point iterations) may diverge if no appropriate initial
values are available.
Moreover, such a purely numerical approach will not find all solutions of the
discretized problem if there are more than one. A final motivation for the use of
symbolic methods
is the fact that all discrete solutions can be obtained as functions of unknown
parameters.
In this paper, we focus on a special class of
partial differential equations derived from geometric problems. A main
challenge in this class is the fact that the polynomial structure of the
nonlinearity is not explicit in the
divergence form usually used for finite element discretization. As a consequence,
the discrete form would always yield some non-polynomial
terms. We therefore consider two different discretizations, namely a polynomial
reformulation before discretization and a direct discretization of the divergence form with
polynomial approximation of the discrete system.
In order to perform a detailed analysis and convergence theory of the discretization
methods we investigate some model problems related to mean-curvature
type equations.
},

number = {05-02},

year = {2005},

month = {May},

note = {accepted to be published in proceeding of SNC'05},

keywords = {Finite element methods, symbolic computation, preprocessing, Newton iteration, multigrid, polynomial equations},

sponsor = {SFB F013/F1304, F1308},

length = {19},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}