Finite-volume-element method for second-order quasilinear elliptic problems

In this paper we consider the finite-volume-element method for general second-order quasilinear elliptic problems over a convex polygonal domain in the plane. Using reasonable assumptions, we show the existence and uniqueness of the finite-volume-element approximations. It is proved that the finite-volume-element approximations are convergent with O(h), O(h(1-2/r) |ln h|), where r > 2, and O(h(2)|ln h|) in the H-1-, W-1,W-infinity- and L-2-norms, respectively, for u is an element of W-2,W-r(Omega) and u is an element of W-2,W-infinity(Omega) boolean AND W-3,W-p(Omega), where p > 1. Moreover, the optimal-order error estimates in the W-1,W-infinity- and L-2-norms and an O(h(2)|ln h|) estimate in the L-infinity-norm are derived under the assumption that u is an element of W-2,W-infinity(Omega) boolean AND H-3(Omega). Numerical experiments are presented to confirm the estimates.