## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 561-575 |

Number of pages | 15 |

Journal | Numerische Mathematik |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1990 |

## Keywords

- Subject Classifications: AMS(MOS): 65N30, 35J60, CR: G1.8

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics
- Mathematics(all)