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Summary  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.

A second‐order face‐centred finite volume method on general meshes with automatic mesh adaptation