Crank-Nicolson
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Optimal Time-Adaptivity for Parabolic Problems
arXiv:2512.05676v2 Announce Type: replace Abstract: Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence...
Well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations
new Abstract: The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfv\'en wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincar\'e-type inequality along the integral curves of $\b$, which holds...