Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems

Abstract

A mesh condition is developed for linear finite element approximations of anisotropic diffusionconvection-reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and O( b ∞ h + c ∞ h 2 )-nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where h denotes the mesh size and b and c are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an O( b ∞ h + c ∞ h 2 ) perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.

On the finite element approximation of elliptic QVIs with noncoercive operators

“…Furthermore, if the matrix ( a j k ) involved in the bilinear form is symmetric ( a j k = a k j ), then mesh conditions for which the off‐diagonal entries of B satisfy. can be found in the previous study . Therefore, combining , and , we have the following lemma.…”

A mixed nonoverlapping covolume method on quadrilateral grids for elliptic problems

“…In the following figures, we denote by ∥u − u h ∥ div , ∥p − p h ∥ 0 , ∥Π h u − u h ∥ 0 and ∥Θ h p − p h ∥ 0 , the error of velocities in the H(div)-norm, error of pressures in the L 2 -norm, superconvergence error of velocities in the L 2 -norm and superconvergence error of pressures in the L 2 -norm, respectively.  , which is anisotropic (see [25]). …”

Preserving nonnegativity of an affine finite element approximation for a convection–diffusion–reaction problem

“…A discontinuous Galerkin method, satisfying a strict maximum principle, was introduced by Zhang, Zhang, and Shu in [13] for a nonlinear convection-diffusion equation. For the case of an anisotropic diffusion-convection-reaction problem, Lu, Huang and Qiu in [8] derived a sufficient condition such that the linear finite element approximation satisfied a discrete maximum principle. In [6], for a coupled system of nonlinear parabolic equations in R 2 , De Leenheer, Gopalakrishnan and Zuhr proposed and analyzed a linear finite element approximation using a backward Euler time discretization.…”

“…In the following, we give a brief summary of recent work on discrete maximum principles for the approximation of elliptic and parabolic differential equations. A detailed description of the development in this area is given in [8,9].…”