Analysis of least-squares approximations to second-order elliptic problems. II. Finite volume method

A theoretical analysis of the combined finite volume/least squares approximations to boundary value problems for second-order elliptic equations in mixed first-order system formulation with variable coefficients in two- and three-dimensional bounded domains is presented. This method is composed of a direct cell vertex finite volume discretization step and an algebraic least-squares step, where the least-squares procedure is applied after the finite volume discretization process is achieved. An optimal error estimate in the H¹(Ω) product norm for continuous piecewise linear approximating function spaces is derived. An equivalent a posteriori error estimator in the H¹(Ω) product norm is also proposed and analyzed.