دانلود رایگان مقاله اپراتورهای بازسازی کم نظم در مش چند وجهی

Abstract

We study low-order reconstruction operators on polyhedral meshes, providing a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. We present two equivalent sets of design properties and draw links with the literature. In particular, the two-level construction based on a P0-consistent and a stabilization part provides a systematic way of designing these operators. We present a simple example of piecewise constant reconstruction in each mesh cell, relying on geometric identities to fulfill the design properties on polyhedral meshes. Finally, we use these reconstruction operators to define a Hodge inner product and build Compatible Discrete Operator schemes, and we test the influence of the reconstruction operators in terms of accuracy and computational efficiency on an anisotropic diffusion problem.

7. Conclusion

In this work, we have studied low-order reconstruction operators for polyhedral meshes in a unified framework for degrees of freedom attached to vertices, edges, faces, and cells. These reconstruction operators provide a systematic way of building a Hodge inner product which is a key concept for the compatible numerical approximation of PDEs. We have presented two equivalent sets of design properties. Moreover, a simple example of piecewise constant reconstruction operators depending on a single stabilization parameter has been detailed, and the influence of this parameter on accuracy and computational costs has been investigated numerically on an anisotropic diffusion problem using CDO vertex-based schemes. Under- and over-penalized values of the stabilization parameter have a negative impact, on the preservation of bounds and on accuracy and costs, respectively. For the problem considered, appropriate choices are values closed to those proposed in DGA and HFV schemes. These conclusions are to be confirmed by further numerical tests.