Steady-state inhomogeneous diffusion with generalized oblique boundary conditions

We consider the elliptic diffusion (steady-state heat conduction) equation with space-dependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well- posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes. We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion $\Gamma_{0}$ of the boundary $\partial \Omega$ from Cauchy data on the complementary portion $\Gamma_{1} = \partial \Omega \backslash \Gamma_{0}$.