Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations

Strongly anisotropic diffusion equations require special techniques to overcome or reduce the mesh locking phenomenon. We present a finite volume scheme that tries to approximate with the best possible accuracy the quantities that are of importance in discretizing anisotropic fluxes. In particular, we discuss the crucial role of accurate evaluations of the tangential components of the gradient acting tangentially to the control volume boundaries, that are called into play by anisotropic diffusion tensors. To obtain the sought characteristics from the proposed finite volume method, we employ a second-order accurate reconstruction scheme which is used to evaluate both normal and tangential cell-interface gradients. The experimental results on a number of different meshes show that the scheme maintains optimal convergence rates in both L^2 and H^1 norms except for the benchmark test considering full Neumann boundary conditions on non-uniform grids. In such a case, a severe locking effect is experienced and documented. However, within the range of practical values of the anisotropy ratio, the scheme is robust and efficient. We postulate and verify experimentally the existence of a quadratic relationship between the anisotropy ratio and the mesh size parameter that guarantees optimal and sub-optimal convergence rates.