Quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes: Optimal-order errors based on Barlow points

This paper presents quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes that use Barlow points (optimal stress points) for dual partitions. Introducing Barlow points into the finite-volume formulations results in better approximation properties at the cost of loss of symmetry. The novel 'symmetrization' technique adopted in this paper allows us to derive optimal-order error estimates in the H1-and L2-norms for elliptic problems and in the L∞ (H1)-and L∞ (L2)-norms for parabolic problems. Superconvergence of the difference between the gradients of the finite-volume solution and the interpolant can also be derived. Numerical results confirm the proved error estimates.