On the Construction and Analysis of High Order Locally Conservative Finite Volume-Type Methods for One-Dimensional Elliptic Problems

Abstract

Locally conservative, finite volume-type methods based on continuous piecewise polynomial functions of degree r \ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension. The new methods extend and generalize the classical finite volume method based on piecewise linear functions. We derive a priori error estimates in the L2 , H1 , and L^\infty norm and discuss superconvergence effects for the error and its derivative. Explicit, residual-based a posteriori error bounds in the L2 and energy norm are also derived. We compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates.