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On Nonnegativity Preservation in Finite Element Methods for the Heat Equation with Non-Dirichlet Boundary Conditions

In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Author

Listed:
  • Stig Larsson

    (Chalmers University of Technology and University of Gothenburg, Department of Mathematical Sciences)

  • Vidar Thomée

    (Chalmers University of Technology and University of Gothenburg, Department of Mathematical Sciences)

Abstract

By the maximum principle the solution of the homogeneous heat equation with homogeneous Dirichlet boundary conditions is nonnegative for positive time if the initial values are nonnegative. In recent work it has been shown that this does not hold for the standard spatially discrete and fully discrete piecewise linear finite element methods. However, for the corresponding semidiscrete and Backward Euler Lumped Mass methods, nonnegativity of initial data is preserved, provided the underlying triangulation is of Delaunay type. In this paper, we study the corresponding problems where the homogeneous Dirichlet boundary conditions are replaced by Neumann and Robin boundary conditions, and show similar results, sometimes requiring more refined technical arguments.

Suggested Citation

  • Stig Larsson & Vidar Thomée, 2018. "On Nonnegativity Preservation in Finite Element Methods for the Heat Equation with Non-Dirichlet Boundary Conditions," Springer Books, in: Josef Dick & Frances Y. Kuo & Henryk Woźniakowski (ed.), Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, pages 793-814, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-72456-0_35
    DOI: 10.1007/978-3-319-72456-0_35
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