Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

A full discretization for the saddle-point approach of a degenerate parabolic problem involving a moving body

“…In this section, we examine the error estimate (13) for some finite element space pairs (V h 0 , V h ), which satisfy the discrete inf-sup condition (8). Let {T h } h>0 be a regular family of triangulations of Ω.…”

“…There are some stable finite element pairs satisfying the discrete inf-sup condition (8), see [13,Section VI.3]. Now, we present two simple examples, namely (P 2 − P 0 ) for discontinuous element space of p and the Taylor-Hood pair (P 2 − P 1 ) for continuous elements.…”

“…Nevertheless, the divergence-free condition and the convection-type term arising from the movement of conductors were not taken into account. The goal of [7] is to present an abstract framework for analyzing a family of linear degenerate parabolic mixed equations, then the paper [8] aims at introducing a fully-discrete approximation for this kind of problems. As stated in [8], the discrete inf-sup condition plays an important role for finite element analysis of the mixed problems, which allowed the authors to get quasi-optimal error estimate O( √ τ + h/ √ τ).…”

“…The goal of [7] is to present an abstract framework for analyzing a family of linear degenerate parabolic mixed equations, then the paper [8] aims at introducing a fully-discrete approximation for this kind of problems. As stated in [8], the discrete inf-sup condition plays an important role for finite element analysis of the mixed problems, which allowed the authors to get quasi-optimal error estimate O( √ τ + h/ √ τ). However, in these articles, all concerned domain and subdomains were fixed during the time process.…”

Finite element error estimates for a mixed degenerate parabolic model

“…Recently, a FEM approximation for the general problem ( 1)-( 3) has been analyzed in [1] and to obtain the convergence of the method has been assumed that u ∈ H 2 (0, T ; Y ). This condition for the concrete case of problems in [3][4][5] is equivalent to suppose that ∂ t t u is square integrable in Ω × (0, T ).…”