A compact finite volume method and its extrapolation for elliptic equations with third boundary conditions

A fourth-order compact finite volume method is constructed for one and two dimensional elliptic equations with third boundary conditions in this paper. Taking two point boundary value problem of third kind as an example, we derive some useful high accuracy post-processing formulas to recover the numerical derivatives at the nodes or midpoints of the elements. We also improve the accuracy of the compact finite volume scheme from order 4 to 6 based on Richardson extrapolation by rigorously proving the scheme has error asymptotic expansion. Numerical examples verify the correctness of the theoretical analysis and also show the effectiveness of the scheme as well as its post-processing formulas and extrapolation.