Analysis of the decoupled and positivity-preserving DDFV schemes for diffusion problems on polygonal meshes

Abstract

We propose and analyze a family of decoupled and positivity-preserving discrete duality finite volume (DDFV) schemes for diffusion problems on general polygonal meshes. These schemes are further extensions of the scheme in Su et al. (J. Comput. Phys.372, 773–798, 2018) and share some benefits. First, the two sets of finite volume (FV) equations are constructed for the vertex-centered and cell-centered unknowns, respectively, and these two sets of equations can be solved in a decoupled way. Second, unlike most existing nonlinear positivity-preserving schemes, nonlinear iteration methods are not required for linear problems and the nonlinear solver can be selected unrestrictedly for nonlinear problems. Moreover, the positivity and well-posedness of the solution can be analyzed rigorously for linear problems. Under some weak geometric assumptions, the stability and error estimate results for the vertex-centered unknowns are obtained. Since the present DDFV schemes and the pure vertex-centered scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016) share the same FV equations for the vertex-centered unknowns, this part itself can also serve as an analysis for the scheme in Wu et al. (Int. J. Numer. Meth. Fluids81(3), 131–150, 2016). Furthermore, by assuming the coercivity of the FV equations for the cell-centered unknowns, a first-order H1 error estimate is obtained for the cell-centered unknowns through the analysis of residual errors. Numerical experiments demonstrate both the accuracy and the positivity of discrete solutions on general grids. The efficiency of the Newton method is emphasized by comparison with the fixed-point method, which illustrates the advantage of our schemes when dealing with nonlinear problems.