L∞ stability of finite element approximations to elliptic gradient equations

Thomas Kerkhoven*, Joseph W Jerome

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

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 languageEnglish (US)
Pages (from-to)561-575
Number of pages15
JournalNumerische Mathematik
Volume57
Issue number1
DOIs
StatePublished - Dec 1 1990

Keywords

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

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Mathematics(all)

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