标题： 关于具有变号系数的PDEs的原始FEM离散化的不稳定性 显示英文标题

标题： On the instabilities of naive FEM discretizations for PDEs with sign-changing coefficients

摘要： 我们考虑一个主部系数为变号的标量扩散方程。 在系数对比非临界的情况下，此类问题的适定性已经得到了广泛研究。 此外，已经提出了许多不同的方法来构建稳定的离散化，因为朴素的有限元离散化通常被认为在一般情况下不可靠。 然而，尚未有明确的实例证明实际的不稳定性，数值实验通常也不以明确的方式表现出不稳定性。 为此，我们构造了一个显式实例，并提供了一个广泛的网格族，我们证明了相应的朴素有限元离散化是不稳定的。 另一方面，我们也提供了一个广泛的（非对称）网格族，我们证明了这些离散化是稳定的。 这两个发现共同解释了数值实验中观察到的结果。 显示英文摘要

摘要： We consider a scalar diffusion equation with a sign-changing coefficient in its principle part. The well-posedness of such problems has already been studied extensively provided that the contrast of the coefficient is non-critical. Furthermore, many different approaches have been proposed to construct stable discretizations thereof, because naive finite element discretizations are expected to be non-reliable in general. However, no explicit example proving the actual instability is known and numerical experiments often do not manifest instabilities in a conclusive manner. To this end we construct an explicit example with a broad family of meshes for which we prove that the corresponding naive finite element discretizations are unstable. On the other hand, we also provide a broad family of (non-symmetric) meshes for which we prove that the discretizations are stable. Together, these two findings explain the results observed in numerical experiments.