Higher-order discrete maximum principle for 1D diffusion-reaction problems

0352122 - MÚ 2011 RIV NL eng J - Journal Article
Vejchodský, Tomáš
Higher-order discrete maximum principle for 1D diffusion-reaction problems.
Applied Numerical Mathematics. Roč. 60, č. 4 (2010), s. 486-500. ISSN 0168-9274. E-ISSN 1873-5460.
[Conference in Numerical Analysis (NumAn 2008). Kalamata, 01.09.2008-05.09.2008]
R&D Projects: GA AV ČR IAA100760702; GA AV ČR(CZ) IAA100190803
Institutional research plan: CEZ:AV0Z10190503
Keywords : discrete maximum principle * discrete Green's function * diffusion-reaction problem * higher-order finite element method * hp-FEM * M-matrix
Subject RIV: BA - General Mathematics
Impact factor: 0.919, year: 2010
http://www.sciencedirect.com/science/article/pii/S0168927409001731

Sufficient conditions for the validity of the discrete maximum principle (DMP) for a 1D diffusion-reaction problem -u '' + kappa(2)u = f with homogeneous Dirichlet boundary conditions discretized by the higher-order finite element method are presented. It is proved that the DMP is satisfied if the lengths h of all elements are shorter then one-third of the length of the entire domain and if kappa(2)h(2) is small enough for all elements. In general, the bounds for kappa(2)h(2) depend on the polynomial degree of the elements, on h, and on the size of the domain. The obtained conditions are simple and easy to verify. A technical assumption (nonnegativity of certain rational functions) was verified by computer for polynomial degrees up to 10. The paper contains an analysis of the discrete Green's function which can be of independent interest.
Permanent Link: http://hdl.handle.net/11104/0191706