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 [13] 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 uh, we compute a Morley-type interpolant Iuh. For the exact solution u, the energy error ‖∇T (u− Iuh)‖L2 can be controlled efficiently and reliably by a residual-based a posteriori error estimator η. The local contributions of η are used to steer an adaptive mesh-refining algorithm. As model example serves the Laplace equation in 2D with mixed Dirichlet-Neumann boundary conditions.