Finite Element Approximation of the p(·)-Laplacian

In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given p € (1, oo), /, and g , find u such that -V • (\Vu\"-2Vu) = f iniîcR2, u = g on a«. The finite element approximation is defined over Í2* , a union of regular triangles, yielding a polygonal approximation to Q. For sufficiently regular solutions u , achievable for a subclass of data /, g , and Í2 , we prove optimal error bounds for this approximation in the norm Wl •Q(Q!1), q = p for p < 2 and q e [ 1, 2] for p > 2, under the additional assumption that Qh Ç £2. Numerical results demonstrating these bounds are also presented.