Discrete maximum principle for FE solutions of the diffusion-reaction problem on prismatic meshes

Abstract

a r t i c l e i n f o MSC: 65N30 65N50 35B50 35J25 Keywords: Diffusion-reaction problem Maximum principle Prismatic finite elements Discrete maximum principle a b s t r a c tIn this paper we analyse the discrete maximum principle (DMP) for a stationary diffusionreaction problem solved by means of prismatic finite elements. We derive geometric conditions on the shape parameters of the prismatic partitions which guarantee validity of the DMP. The presented numerical tests show the sharpness of the obtained conditions.

Discrete maximum principle for parabolic problems solved by prismatic finite elements

“…In [14] we also present explicit expressions for the off-diagonal entries of these matrices. In the parabolic case we will need, in addition, explicit expressions for the diagonal entries, cf.…”

“…Therefore, we omit them here. (1) and (2), combined with the θ-method in the time discretization, satisfies (14) (and, therefore, the DMP (13)) if and only if the conditions: (1) and (2), combined with the θ-method in the time discretization, satisfies the discrete maximum principle (14) if the conditions: …”

“…To utilize the analysis of the elliptic case from [14] as much as possible, we introduce the following theorem. with S (P) given by (15) and with…”

“…Recently we proved the DMP for an elliptic problem discretized by prismatic elements, see [14], where we already computed the entries of the matrix S (P) as well as the entries of the matrices M (n,θ),(P) and C (P) (t) in case (n,θ) …”

“…Let ϕ A (x, y, z) = λ A (x, y) 0 (z) be the shape function corresponding to the vertex A of the prism P, where λ A is the barycentric coordinate corresponding to the vertex A of the base triangle T and 0 (z) = 1 − z/d, z ∈ I, is the 1D shape function. We may easily compute, see [14], the following integrals:…”