Non-negative mixed finite element formulations for a tensorial diffusion equation

K. B. Nakshatrala, A. J. Valocchi

Research output: Contribution to journalArticlepeer-review


We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart-Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum-minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

Original languageEnglish (US)
Pages (from-to)6726-6752
Number of pages27
JournalJournal of Computational Physics
Issue number18
StatePublished - Oct 1 2009


  • Active set strategy
  • Convex quadratic programming
  • Discrete maximum-minimum principle
  • Maximum-minimum principles for elliptic PDEs
  • Monotone methods
  • Non-negative solutions
  • Tensorial diffusion equation

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'Non-negative mixed finite element formulations for a tensorial diffusion equation'. Together they form a unique fingerprint.

Cite this