Finite Element Approximation of the Parabolic p-Laplacian

Abstract

The authors analyse the semidiscrete approximation and a fully discrete approximation using the backward Euler time discretisation, obtaining error bounds which improve on those in the literature.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

“…For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8. Alternative numerical approaches have also been investigated; here we only mention finite element schemes (see [36,16] and references therein), kinetic schemes (see [14,22,55] and references therein) and operator splitting schemes (see [43]). …”

Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities

“…For example k{t) = t p~2 , p e (1, oo), satisfies the Assumptions (A) with a l = a 2 = 1. These improved inequalities are absolutely essential in establishing sharp error bounds for the finite element approximation of some degenerate quasilinear problems for which there is no associated minimization problem (for example, the parabolic p-Laplacian, see [4] …”

“…We note that there is little pomt m considermg a higher order approximation due to the lack of regulanty of the solutions of (LI) in gênerai. It has been further shown that the techniques used m dealmg with (11) can also be applied in a modified form to the case of a quasi-Newtoman flow obeymg the power law or the Carreau law, see [3], and the parabolic p-Laplacian, see [4].…”

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

“…Let us close the introduction by putting our results in contrast to existing FE approximation results of single scale parabolic monotone problems. In the L p (W 1,p ) setting, optimal explicit convergence rates in terms of the discretization parameters have been derived for maps with a p-structure 3 , e.g., the parabolic p-Laplacian, using quasi-norms in space, see [11,20]. Note however that under the assumptions (A 1−2 ) the maps A ε have p-structure if and only if α = 1 and β = 2.…”

“…As we assume 0 < α ≤ min{p − 1, 1} and max{2, p} ≤ β < ∞ (the most general assumptions on the oscillatory maps allowing for homogenization, see [15,18,42]), we have in addition that p = 2 if we want both a p-structure and a valid homogenization setting. For this set of parameters, the quasi-norm (in space) from [11,20] collapses to the standard H 1 (Ω) norm. For all other values of p, homogenization theory seems not to exist for maps A ε with p-structure and thus studying numerical homogenization methods makes no sense.…”