A Posteriori Error Estimate and Adaptive Mesh Refinement for the Cell-Centered Finite Volume Method for Elliptic Boundary Value Problems
Abstract
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.
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Copyright © 2008 Society for Industrial and Applied Mathematics.
History
Submitted: 6 September 2007
Accepted: 9 June 2008
Published online: 24 October 2008
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