A posteriori error analysis for a cut cell finite volume method

We study the solution of a diffusive process in a domain where the diffusion coefficient changes discontinuously across a curved interface. We consider discretizations that use regularly-shaped meshes, so that the interface “cuts” through the cells (elements or volumes) without respecting the regular geometry of the mesh. Consequently, the discontinuity in the diffusion coefficients has a strong impact on the accuracy and convergence of the numerical method. This motivates the derivation of computational error estimates that yield accurate estimates for specified quantities of interest. For this purpose, we adapt the well-known adjoint based a posteriori error analysis technique used for finite element methods. In order to employ this method, we describe a systematic approach to discretizing a cut-cell problem that handles complex geometry in the interface in a natural fashion yet reduces to the well-known Ghost Fluid Method in simple cases. We test the accuracy of the estimates in a series of examples.