FINITE-ELEMENT APPROXIMATION OF THE P-LAPLACIAN

In this paper we consider the continuous piecewise linear finite eleme nt 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 ele ment approximation is defined over OMEGA(h), a union of regular triang les, yielding a polygonal approximation to OMEGA. For sufficiently reg ular solutions u, achievable for a subclass of data f, g, and OMEGA, w e 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, un der the additional assumption that OMEGA(h) subset-or-equal-to OMEGA. Numerical results demonstrating these bounds are also presented.