Finite Difference Schemes on Triangular Cell-Centered Grids with Local Refinement

Based on approximation of the balance equation, finite difference schemes on triangular cell-centered grids are derived. A priori estimates and error analyses are provided. For certain regular triangulations, convergence in the discrete -norm is established. Also, finite difference schemes on triangular cell-centered grids with local refinement are derived with accuracy , where for a simple symmetric scheme, for a nonsymmetric and a more accurate symmetric scheme, and for a more accurate nonsymmetric scheme. Certain algebraic properties of the corresponding matrices of the derived finite difference schemes are verified, thus allowing the recently proposed algebraic theory for the Bramble–Ewing–Pasciak–Schatz (BEPS) and Fact Adaptive Composite (FAC) two-grid preconditioners to apply. Numerical experiments that demonstrate the accuracy of the difference schemes and the fast convergence of the two-grid BEPS and FAC preconditioners in conjugate gradient-type iterative methods are presented.