A Posteriori Error Estimate and Adaptive Mesh Refinement for the Cell-Centered Finite Volume Method for Elliptic Boundary Value Problems

We extend a result of Nicaise [SIAM J. Numer. Anal., 43 (2005), pp. 1481-1503] for the a posteriori error estimation of the cell-centered finite volume method for the numerical solution of elliptic problems. Having computed the piecewise constant finite volume solution $u_h$, we compute a Morley-type interpolant $\mathcal{I} u_h$. For the exact solution $u$, the energy error $\norm{\nabla_{\mathcal{T}}(u-\mathcal{I} u_h)}{L^2}$ can be controlled efficiently and reliably by a residual-based a posteriori error estimator $\eta$. The local contributions of $\eta$ are used to steer an adaptive mesh-refining algorithm. A model example serves the Laplace equation in two dimensions with mixed Dirichlet-Neumann boundary conditions.