We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which we deduce conservative flux instead of directly prescribing a flux formula and solving the usual flux continuity equations. The boundary conditions of our local variational problems are handled through additional cell-centered unknowns, leading to an overall scheme with the same number of unknowns than first-order discontinuous Galerkin methods. Convergence results follow from well-established frameworks, while numerical experiments illustrate the good behavior of the method.

中文翻译：

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无条件稳定的小模板丰富了一般网格上异质扩散问题的多点通量近似

我们针对一般网格上维度 2 和 3 的异质和各向异性扩散问题的有限体积近似推导了新的多点通量近似 (MPFA)。所得方法是无条件稳定的，同时保留了 MPFA 有限体积典型的小模板。关键思想是通过精心设计的稳定项来解决局部变分问题，从中推导出保守通量，而不是直接规定通量公式并求解通常的通量连续性方程。我们的局部变分问题的边界条件是通过附加的以单元为中心的未知数来处理的，从而产生与一阶不连续伽辽金方法具有相同数量的未知数的总体方案。收敛结果遵循完善的框架，而数值实验则说明了该方法的良好行为。