Correction of Reconstruction Stencils in Finite Volume Scheme for Laplace Equation

For Laplace's equation, we state a control volume, which guarantees a positive finite volume scheme with linear reconstruction of the solution on any unstructured grid. The control volume is defined by a property of the analytical solution to the equation and does not depend on grid geometry. For those problems where the choice of the control volume is prescribed a priori, we discuss how to improve positivity of the finite volume scheme with the linear reconstruction by using corrected reconstruction stencils. Numerical examples illustrated the developed approach to the stencil correction are considered