L _∞ stability of finite element approximations to elliptic gradient equations

Summary

We examine theL _∞ stability of piecewise linear finite element approximationsU to the solutionu to elliptic gradient equations of the form −∇·[a(x)∇u]+f(x, u)=g(x) wheref is monotonically increasing inu. We identify a prioriL _∞ bounds for the finite element solutionU, 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 thatf 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 thatU is stable inL _∞ in two dimensions for a discretization mesh on which −∇·[a(x)∇u] gives rise to anM-matrix, whileU is stable for any mesh in one dimension. The condition that the discretization of −∇·[a(x)∇u] has to be anM-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 forU are obtained by use of (nonexpansive) pointwise projection operators.