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 It can be viewed here in the following format: