Numerical analysis of a mixed elliptic problem with flux and convective boundary conditions to obtain a discrete solution of nonconstant sign
Domingo Alberto Tarzia · Numerical Methods for Partial Differential Equations · 1999
We consider a material that occupies a convex polygonal bounded domain Ω ⊂ ℝn, with regular boundary Γ = Γ1 ∪ Γ2 (with Γ ∩ Γ = ∅︁) with meas (Γ1) = |Γ1| > 0 and |Γ2| > 0. We assume, without loss of generality, that the melting temperature is 0°C. We consider the following steady-state heat conduction problem in Ω: with α, q, B = Const > 0, and q and α represent the heat flux on Γ2 and the heat transfer coefficient on Γ1, respectively. In a previous article (Tabacman- Tarzia, J Diff Eq 77 (1989), 16– 37) sufficient and/or necessary conditions on data α, q, B, Ω, Γ1, Γ2 to obtain a temperature u of nonconstant sign in Ω (that is, a multidimensional steady-state, two-phase, Stefan problem) were studied. In this article, we consider a regular triangulation by finite element method of the domain Ω with Lagrange triangles of the type 1, with h > 0 the parameter of the discretization. We study sufficient (and/or necessary) conditions on data α, q, B, Ω, Γ1, and Γ2 to obtain a change of phase (steady-state, two-phase, discretized Stefan problem) in corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Ω. Moreover, error bounds as a function of the parameter h, are also obtained. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq. 15: 355–369, 1999