Nonlinear finite elements: Sub- and supersolutions for the heterogeneous logistic equation Articles uri icon

publication date

  • March 2021

start page

  • 189

end page

  • 219

volume

  • 278

International Standard Serial Number (ISSN)

  • 0022-0396

Electronic International Standard Serial Number (EISSN)

  • 1090-2732

abstract

  • In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

keywords

  • discrete maximum principle; logistic equation; newton method; nonlinear finite element method; nonlinear galerkin method; subsolutions and supersolutions