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

Abstract

In this paper we study the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in Rd with d=l or 2. Given T>0, f and uO; find u∊K, where K is a closed convex subset of the So bo lev space , such that for x∊Ω and for any v∊K for a.e. t∊(0,T], where k∊C(0,∞) is a given nonnegative function with k(s)s strictly increasing for s≥O, but possibly degenerate, and p∊(1,∞) depends on k. For such a general problem we establish error bounds in energy type norms for a fully discrete approximation based on the backward Euler time discretisation. We show that these error bounds converge at the optimal rate with respect to the space discretisation, provided p≤2 and the solution u is sufficiently regular.