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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.

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