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.

中文翻译：

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解耦和保性的DDFV方案用于多边形网格上的扩散问题

我们提出并分析了用于一般多边形网格上扩散问题的一系列解耦且保持正性的离散对偶有限体积（DDFV）方案。这些方案是Su等人的方案的进一步扩展。（J.计算机物理372（773–798，2018年）并分享一些收益。首先，分别为以顶点为中心和以单元为中心的未知数构造两组有限体积（FV）方程，并且可以以解耦的方式求解这两组方程。其次，与大多数现有的非线性正性保存方案不同，线性问题不需要非线性迭代方法，并且非线性问题可以不受限制地选择非线性求解器。此外，可以针对线性问题严格分析解决方案的正性和适定性。在一些弱几何假设下，获得了以顶点为中心的未知数的稳定性和误差估计结果。由于目前的DDFV方案和Wu等人的以顶点为中心的纯方案。（Int。J. Numer。Meth。Fluids 81（3），131–150，2016）对以顶点为中心的未知数共享相同的FV方程，这部分本身也可以用作Wu等人对该方案的分析。（Int。J.Numer.Meth.Fluids 81（3），131-150，2016）。此外，通过假设以单元为中心的未知数的FV方程的矫顽力，可以通过分析残差来获得以单元为中心的未知数的一阶H ¹误差估计。数值实验证明了一般网格上离散解的准确性和正性。通过与定点方法进行比较来强调牛顿方法的效率，这说明了我们的方法在处理非线性问题时的优势。