Discrete maximum principle for linear parabolic problems solved on hybrid meshes

Discrete maximum principles for FE solutions of nonstationary diffusion-reaction problems with mixed boundary conditions

“…• For the particular case c = 0 and σ = 0, the final conditions (54) and (55) reduce to some known [11,10] requirements for DMPs.…”

“…If the finite element method (FEM) is employed, the corresponding DMPs are normally ensured by imposing certain geometrical restrictions on the spatial meshes used: such as acuteness or nonobtuseness -for simplicial meshes [6,11,13,16], non-narrowness -for rectangular meshes [3,9,12]. In [10], sufficient geometric conditions for DMPs are given for the case of planar hybrid meshes. The validity of DMPs for higher order finite elements and associated geometric restrictions on FE meshes are considered in [23].…”

A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems

“…In the case of linear and bilinear finite element approximations of the Poisson equation, they are satisfied automatically on a suitably designed mesh (triangles of acute/nonobtuse type, quadrilaterals of nonnarrow type [6,12,13]). However, the involved geometric constraints turn out too restrictive in the case of problem (7) with an anisotropic diffusion tensor.…”

“…As before, the source term is zero. The computational domain Ω = (0, 1) 2 \(Ω 4,6 ∪Ω 8,6 ) has two square holes that correspond to cells (4,6) and (8,6) of a uniform grid with 11 × 11 cells. The Dirichlet boundary conditions prescribed on Γ 1 = ∂Ω 4,6 and Γ 2 = ∂Ω 8,6 are as follows…”

“…For a triangulation of acute or nonobtuse type (all angles smaller than or equal to π/2) the piecewise-linear finite element approximation of the Poisson equation produces a discrete operator that proves to be an M-matrix [4,12,13]. In the case of bilinear finite elements, it is sufficient to require that all quadrilaterals be of nonnarrow type (aspect ratios smaller than or equal to √ 2 [6]). However, the geometric approach to DMP verification fails in the case of higherorder finite elements [9], singularly perturbed convection-diffusion equations [15], and anisotropic diffusion problems [17].…”