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 is-an-element-of (1, infinity), f, and g , find u such that -del . (\delu\p-2delu) = f in OMEGA subset-of R2, u = g on partial derivative OMEGA. The finite element approximation is defined over OMEGA(h), a union of regular triangles, yielding a polygonal approximation to OMEGA. For sufficiently regular solutions u, achievable for a subclass of data f, g, and OMEGA, we prove optimal error bounds for this approximation in the norm W1,q (OMEGA(h)) , q = p for p < 2 and q is-an-element-of [1,2] for p > 2, under the additional assumption that OMEGA(h) subset-or-equal-to OMEGA. Numerical results demonstrating these bounds are also presented.