A second-order face-centred finite volume strategy on general meshes is proposed. The method uses a mixed formulation in which a constant approximation of the unknown is computed on the faces of the mesh. Such information is then used to solve a set of problems, independent cell-by-cell, to retrieve the local values of the solution and its gradient. The main novelty of this approach is the introduction of a new basis function, utilised for the linear approximation of the primal variable in each cell. Contrary to the commonly used nodal basis, the proposed basis is suitable for computations on general meshes, including meshes with different cell types. The resulting approach provides second-order accuracy for the solution and first-order for its gradient, without the need of reconstruction procedures, is robust in the incompressible limit and insensitive to cell distortion and stretching. The second-order accuracy of the solution is exploited to devise an automatic mesh adaptivity strategy. An efficient error indicator is obtained from the computation of one extra local problem, independent cell-by-cell, and is used to drive mesh adaptivity. Numerical examples illustrating the approximation properties of the method and of the mesh adaptivity procedure are presented. The potential of the proposed method with automatic mesh adaptation is demonstrated in the context of microfluidics.

中文翻译：

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具有自动网格自适应的一般网格的二阶面心有限体积方法

提出了一般网格上的二阶面心有限体积策略。该方法使用混合公式，其中在网格的面上计算未知数的常数近似值。然后使用这些信息逐个独立地解决一组问题，以检索解决方案的局部值及其梯度。这种方法的主要新颖之处在于引入了一个新的基函数，用于每个单元格中原始变量的线性近似。与常用的节点基相反，所提出的基适用于一般网格的计算，包括具有不同单元类型的网格。由此产生的方法为解提供二阶精度，为其梯度提供一阶精度，无需重建程序，在不可压缩极限中具有鲁棒性，并且对单元变形和拉伸不敏感。利用解的二阶精度来设计自动网格自适应策略。一个有效的误差指标是从一个额外的局部问题的计算中获得的，逐个独立的单元，并用于驱动网格自适应。给出了说明该方法和网格自适应程序的近似特性的数值示例。在微流体的背景下证明了所提出的具有自动网格自适应的方法的潜力。并用于驱动网格自适应。给出了说明该方法和网格自适应程序的近似特性的数值示例。在微流体的背景下证明了所提出的具有自动网格自适应的方法的潜力。并用于驱动网格自适应。给出了说明该方法和网格自适应程序的近似特性的数值示例。在微流体的背景下证明了所提出的具有自动网格自适应的方法的潜力。