We examine the L∞ stability of piecewise linear finite element approximations U to the solution u to elliptic gradient equations of the form -∇·[a(x)∇u]+f(x, u)=g(x) where f is monotonically increasing in u. We identify a priori L∞ bounds for the finite element solution U, which we call "reduced" bounds, and which are marginally weaker than those for the original differential equations. For the general, N-dimensionai, case we identify new conditions on the mesh, such that under the assumption that f is Lipschitz continuous on a finite interval, U satisfies the "reduced"L∞ bounds mentioned above. The new, N-dimensional regularity conditions preclude quasi-rectangular meshes. Moreover, we show that U is stable in L∞ in two dimensions for a discretization mesh on which -∇·[a(x)∇u] gives rise to an M-matrix, while U is stable for any mesh in one dimension. The condition that the discretization of -∇·[a(x)∇u] has to be an M-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion. The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions for U are obtained by use of (nonexpansive) pointwise projection operators.
- Subject Classifications: AMS(MOS): 65N30, 35J60, CR: G1.8
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics