In this paper, we propose a non-orthodox Multipoint Flux Approximation scheme with a “Diamond” stencil (MPFA-D) for the solution of the 3-D steady state diffusion equation. Following the work of GAO and WU (2011), in our method, the auxiliary vertex unknowns are eliminated by a novel explicit interpolation that is flux conservative and is constructed under the Linearity-Preserving Criterion (LPC). The MPFA-D is able to reproduce piecewise linear solutions exactly on challenging heterogeneous and anisotropic media, even in cases with some severely distorted meshes. Furthermore, our new scheme presentssecond order accuracy for the scalar unknown and, at least, first order accuracy for fluxes, considering unstructured tetrahedral meshes and arbitrarily anisotropic diffusion tensors. In order to validate our numerical scheme, we perform different test cases, involving 3-D benchmarks on diffusion problems. We compare the performance with other schemes found in literature. We also compare our Linearity-Preserving Explicit Weight (LPEW) interpolation with other interpolations strategies to evaluate its robustness to handle anisotropic and heterogeneous, possibly discontinuous diffusion tensors. In general, our linear preserving MPFA-D method performs well, however it is not monotone, particularly for very distorted meshes and highly anisotropic diffusion tensors.

在本文中，我们针对3-D稳态扩散方程的求解提出了一种非常规的多点通量近似方案，该方案采用“菱形”模版（MPFA-D）。遵循GAO和WU（2011）的工作，在我们的方法中，通过新颖的显式插值消除了辅助顶点未知量，该显式插值是通量保守的，并根据线性保留准则（LPC）构造。MPFA-D能够精确地在具有挑战性的非均质和各向异性介质上重现分段线性解，即使在某些网格严重变形的情况下也是如此。此外，考虑到非结构化的四面体网格和任意各向异性的扩散张量，我们的新方案对标量未知数表示二阶精度，对于通量至少表示一阶精度。为了验证我们的数值方案，我们执行不同的测试案例，涉及有关扩散问题的3-D基准测试。我们将性能与文献中发现的其他方案进行比较。我们还将保持线性的显式权重（LPEW）插值与其他插值策略进行比较，以评估其处理各向异性和异质（可能是不连续的）扩散张量的鲁棒性。通常，我们的线性保留MPFA-D方法表现良好，但是它不是单调的，尤其是对于变形非常严重的网格和高度各向异性的扩散张量而言。可能是不连续的扩散张量。通常，我们的线性保留MPFA-D方法表现良好，但是它不是单调的，尤其是对于变形非常严重的网格和高度各向异性的扩散张量而言。可能是不连续的扩散张量。通常，我们的线性保留MPFA-D方法表现良好，但是它不是单调的，尤其是对于变形非常严重的网格和高度各向异性的扩散张量而言。