A Trapping Principle and Convergence Result for
Finite Element Approximate Solutions
of Steady Reaction/Diffusion Systems
By: Joseph W. Jerome
We consider nonlinear elliptic systems, with mixed boundary
conditions,
on a convex polyhedral domain in Euclidean space.
These are nonlinear divergence form systems, with vector field f
outward pointing on the
trapping region boundary.
The motivation is that of
applications to steady-state reaction/diffusion systems.
Also included are reaction/diffusion/convection systems which satisfy the
Einstein relations, for which the Cole-Hopf transformation is possible.
For maximum generality, the theory is not tied to any specific
application.
We are able to demonstrate
a trapping principle
for the piecewise linear Galerkin approximation, defined via
a lumped integration
hypothesis on integrals involving f, by use of variational
inequalities.
Results of this type have previously been obtained for parabolic systems
by Estep, Larson, and Williams, and for nonlinear elliptic equations by
Karatson
and Korotov. Recent minimum and maximum principles have
been obtained by Juengel and Unterreiter for nonlinear
elliptic equations.
We make use of special properties of the element stiffness
matrices, induced by a geometric constraint upon the simplicial
decomposition. This constraint is known as the non-obtuseness condition.
It states that the inward normals, associated with an arbitrary pair
of an element's faces, determine an angle with nonpositive cosine.
Draganescu,
Dupont, and Scott have constructed an example for which the discrete
maximum principle fails if this condition is omitted.
We also assume vertex communication in each element
in the form of an
irreducibility hypothesis on the off-diagonal elements of the stiffness
matrix.
There is a
companion convergence result, which yields an existence theorem for the
solution.
This entails a consistency
hypothesis for interpolation on the boundary,
and depends on the Tabata construction of simple function
approximation,
based on barycentric regions. This paper has appeared in Numerische
Mathematik: vol. 109 (2008), 121--142.
For access, go to www.springerlink.com. The article's DOI is:
DOI:10.1007/s00211-008-0136-z
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